Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) [English] 11 Standard Maharashtra State Board chapter 4 - Determinants and Matrices [Latest edition]

Find the value of determinant : 

`|(2, -4),(7, -15)|`

Find the value of determinant :

`|(2"i", 3),(4, -"i")|`

Find the value of determinant :

`|(3, -4, 5),(1, 1, -2),(2, 3, 1)|`

Find the value of determinant :

`|("a", "h", "g"),("h", "b", "f"),("g", "f", "c")|`

Find the value of x if

`|(x^2 - x + 1, x + 1),(x + 1, x + 1)|` = 0

Find the value of x if `|(x, -1, 2),(2x, 1, -3),(3, -4, 5)|` = 29 

Find x and y if `|(4"i", "i"^3, 2"i"),(1, 3"i"^2, 4),(5, -3, "i")|` = x + iy where i² = – 1

Find the minor and cofactor of element of the determinant

D = `|(2, -1, 3),(1, 2, -1),(5, 7, 2)|`

Evaluate A = `|(2, -3,5),(6, 0, 4),(1, 5, -7)|` Also find minor and cofactor of elements in the 2^nd row of determinant and verify − a₂₁.M₂₁ + a₂₂.M₂₂ − a₂₃.M₂₃ = value of A

where M₂₁, M₂₂ , M₂₃ are minor of a₂₁ , a₂₂, a₂₃ and C₂₁, C₂₂, C₂₃ are cofactor of a₂₁, a₂₂, a₂₃

Evaluate A = `|(2, -3,5),(6, 0, 4),(1, 5, -7)|` Also find minor and cofactor of elements in the 2^nd row of determinant and verify a₂₁ C₂₁ + a₂₂ C₂₂ + a₂₃ C₂₃ = value of A

where M₂₁, M₂₂, M₂₃ are minors of a₂₁, a₂₂, a₂₃ and

C₂₁, C₂₂, C₂₃ are cofactors of a₂₁, a₂₂, a₂₃.

Find the value of determinant expanding along third column

`|(-1, 1, 2),(-2, 3, -4),(-3, 4, 0)|`

Without expanding evaluate the following determinant:

`|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`

Without expanding evaluate the following determinant:

`|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|`

Without expanding evaluate the following determinant:

`|(2, 7, 65),(3, 8, 75),(5, 9, 86)|`

Prove that `|(x + y, y + z, z + x),(z + x, x + y, y + z),(y + z, z + x, x + y)| = 2|(x, y, z),(z, x, y),(y, z, x)|`

Using properties of determinant show that

`|("a" + "b", "a", "b"),("a", "a" + "c", "c"),("b", "c", "b" + "c")|` = 4abc

Using properties of determinant show that

`|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` = 0

Solve the following equation: 

`|(x + 2, x + 6, x - 1),(x + 6, x - 1, x + 2),(x - 1, x + 2, x + 6)|` = 0

Evaluate the following determinants:

`|(x - 1, x, x - 2),(0, x - 2, x - 3),(0, 0, x - 3)| = 0`

If  `|(4 + x, 4 - x, 4 - x),(4 - x,4 + x,4 - x),(4 - x,4 - x, 4 + x)|` = 0, then find the values of x.

Without expanding determinants show that

`|(1, 3, 6),(6, 1, 4),(3, 7, 12)| + 4|(2, 3, 3),(2, 1, 2),(1, 7, 6)| = 10|(1, 2, 1),(3, 1, 7),(3, 2, 6)|`

Solve the following linear equations by using Cramer’s Rule:

x + y + z = 6, x – y + z = 2, x + 2y – z = 2

Solve the following linear equations by using Cramer’s Rule:

x + y − 2z = –10, 2x + y – 3z = –19, 4x + 6y + z = 2

Solve the following linear equations by using Cramer’s Rule:

x + z = 1, y + z = 1, x + y = 4

Solve the following linear equations by using Cramer’s Rule:

`(-2)/x - 1/y - 3/z = 3, 2/x - 3/y + 1/z = -13 and 2/x - 3/z` = – 11

The sum of three numbers is 15. If the second number is subtracted from the sum of first and third numbers then we get 5. When the third number is subtracted from the sum of twice the first number and the second number, we get 4. Find the three numbers.

Examine the consistency of the following equation:

2x − y + 3 = 0, 3x + y − 2 = 0, 11x + 2y − 3 = 0

Examine the consistency of the following equation:

2x + 3y − 4 = 0, x + 2y = 3, 3x + 4y + 5 = 0

Examine the consistency of the following equation:

x + 2y −3 = 0, 7x + 4y − 11 = 0, 2x + 4y − 6 = 0

Find k if the following equations are consistent:

2x + 3y - 2 = 0, 2x + 4y − k = 0, x − 2y + 3k =0

Find k if the following equations are consistent:

kx + 3y + 1 = 0, x + 2y + 1 = 0, x + y = 0

Find the area of triangle whose vertices are

A(5, 8), B(5, 0) C(1, 0)

Find the area of triangle whose vertices are

`"P"(3/2, 1), "Q"(4, 2), "R"(4, (-1)/2)`

Find the area of triangle whose vertices are

M(0, 5), N(−2, 3), T(1, −4)

Find the area of quadrilateral whose vertices are 

A(−3, 1), B(−2, −2), C(3, -1), D(1, 4)

Find the value of k, if the area of triangle whose vertices are P(k, 0), Q(2, 2), R(4, 3) is `3/2 "sq.unit"`

Examine the collinearity of the following set of point:

A(3, −1), B(0, −3), C(12, 5)

Examine the collinearity of the following set of point:

P(3, −5), Q(6, 1), R(4, 2)

Examine the collinearity of the following set of point:

`"L"(0, 1/2), "M"(2, -1), "N"(-4, 7/2)`

Select the correct option from the given alternatives:

The determinant D = `|("a", "b", "a" + "b"),("b", "c", "b" + "c"),("a" + "b", "b" + "c", 0)|` = 0 if

If `|("x"^"k", "x"^("k" + 2), "x"^("k" + 3)),("y"^"k", "y"^("k" + 2), "y"^("k" + 3)),("z"^"k", "z"^("k" + 2), "z"^("k" + 3))|` = (x - y) (y - z) (z - x)`(1/"x"+ 1/"y" + 1/"z") ` then

Select the correct option from the given alternatives:

Let D = `|(sintheta*cosphi, sintheta*sinphi, costheta),(costheta*cosphi, costheta*sinphi, -sintheta),(-sintheta*sinphi, sintheta*cosphi, 0)|` then

Select the correct option from the given alternatives:

The value of a for which system of equation a³x + (a + 1)³ y + (a + 2)³z = 0 ax + (a +1)y + (a + 2)z = 0 and x + y + z = 0 has non zero Soln. is

Select the correct option from the given alternatives:

`|("b" + "c", "c" + "a", "a" + "b"),("q" + "r", "r" + "p", "p" + "q"),(y + z, z + x, x + y)|` = 

Select the correct option from the given alternatives:

The system 3x – y + 4z = 3, x + 2y – 3z = –2 and 6x + 5y + λz = –3 has at least one Solution when

Select the correct option from the given alternatives:

If x = –9 is a root of `|(x, 3, 7),(2, x, 2),(7, 6, x)|` = 0 has other two roots are

Select the correct option from the given alternatives:

If `|(6"i", -3"i", 1),(4, 3"i", -1),(20, 3, "i")|` = x + iy then

Select the correct option from the given alternatives:

If A(0,0), B(1,3) and C(k,0) are vertices of triangle ABC whose area is 3 sq.units then value of k is

Select the correct option from the given alternatives:

Which of the following is correct

Answer the following question:

Evaluate `|(2, -5, 7),(5, 2, 1),(9, 0, 2)|`

Answer the following question:

Evaluate `|(1, -3, 12),(0, 2, -4),(9, 7, 2)|`

Answer the following question:

Evaluate determinant along second column

`|(1, -1, 2),(3, 2, -2),(0, 1, -2)|`

Answer the following question:

Evaluate `|(2, 3, 5),(400, 600, 1000),(48, 47, 18)|` by using properties

Answer the following question:

Evaluate `|(101, 102, 103),(106, 107, 108),(1, 2, 3)|` by using properties

Answer the following question:

Find minor and cofactor of elements of the determinant:

`|(-1, 0, 4),(-2, 1, 3),(0, -4, 2)|`

Answer the following question:

Find minor and cofactor of elements of the determinant:

`|(1, -1, 2),(3, 0, -2),(1, 0, 3)|`

Answer the following question:

Find the value of x if

`|(1, 4, 20),(1, -2, -5),(1, 2x, 5x^2)|` = 0

Answer the following question:

Find the value of x if

`|(1, 2x, 4x),(1, 4, 16),(1, 1, 1)| = 0`

Answer the following question:

By using properties of determinant prove that `|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|` = 0

Answer the following question:

Without expanding determinant show that

`|("b" + "c", "bc", "b"^2"c"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)|` = 0

Answer the following question:

Without expanding determinant show that

`|(x"a", y"b", z"c"),("a"^2, "b"^2, "c"^2),(1, 1, 1)| = |(x, y, z),("a", "b", "c"),("bc", "ca", "ab")|`

Answer the following question:

Without expanding determinant show that

`|(l, "m", "n"),("e", "d", "f"),("u", "v", "w")| = |("n", "f", "w"),(l, "e", "u"),("m", "d", "v")|`

Answer the following question:

Without expanding determinant show that

`|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)|` = 0

Answer the following question:

If `|("a", 1, 1),(1, "b", 1),(1, 1, "c")|` = 0 then show that `1/(1 - "a") + 1/(1 - "b") + 1/(1 - "c")` = 1

Answer the following question:

Solve the following linear equations by Cramer’s Rule:

2x − y + z = 1, x + 2y + 3z = 8, 3x + y − 4z =1

Answer the following question:

Solve the following linear equations by Cramer’s Rule:

`1/x + 1/y = 3/2, 1/y + 1/z = 5/6, 1/z + 1/x = 4/3`

Answer the following question:

Solve the following linear equations by Cramer’s Rule:

2x+ 3y + 3z = 5 , x − 2y + z = – 4 , 3x – y – 2z = 3

Answer the following question:

Solve the following linear equations by Cramer’s Rule:

x – y + 2z = 7 , 3x + 4y – 5z = 5 , 2x – y + 3z = 12

Answer the following question:

Find the value of k, if the following equations are consistent:

(k + 1)x + (k – 1)y + (k – 1) = 0
(k – 1)x + (k + 1)y + (k – 1) = 0
(k – 1)x + (k – 1)y + (k + 1) = 0

Answer the following question:

Find the value of k, if the following equations are consistent:

3x + y − 2 = 0 kx + 2y − 3 = 0 and 2x − y = 3

Answer the following question:

Find the value of k if the following equation are consistent:

(k − 2)x + (k − 1)y = 17 , (k − 1)x + (k − 2)y = 18 and x + y = 5

Answer the following question:

Find the area of triangle whose vertices are A(−1, 2), B(2, 4), C(0, 0)

Answer the following question:

Find the area of triangle whose vertices are P(3, 6), Q(−1, 3), R(2, −1)

Answer the following question:

Find the area of triangle whose vertices are L(1, 1), M(−2, 2), N(5, 4)

Answer the following question:

Find the value of k:

If area of triangle is 4 square unit and vertices are P(k, 0), Q(4, 0), R(0, 2)

Answer the following question:

Find the value of k:

If area of triangle is `33/2` square unit and vertices are L(3, −5), M(−2, k), N(1, 4)

Answer the following question:

Find the area of quadrilateral whose vertices are A(0, −4), B(4, 0), C(−4, 0), D(0, 4)

Answer the following question:

An amount of ₹ 5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is ₹ 350. If the combined income from the first two investments is ₹ 70 more than the income from the third. Find the amount of each investment.

Answer the following question:

Show that the lines x − y = 6, 4x − 3y = 20 and 6x + 5y + 8 = 0 are concurrent. Also find the point of concurrence

Answer the following question:

Show that the following points are collinear by determinant:

L(2,5), M(5,7), N(8,9)

Answer the following question:

Show that the following points are collinear by determinant:

P(5,1), Q(1,−1), R(11,4)

Construct a matrix A = [a_ij]_{3 × 2} whose element a_ij is given by

a_ij = `(("i" - "j")^2)/(5 - "i")`

Construct a matrix A = [a_ij]_{3 × 2} whose element a_ij is given by

a_ij = i – 3j

Construct a matrix A = [a_ij]_{3 × 2} whose element a_ij is given by

a_ij = `(("i" + "j")^3)/5`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(3, -2, 4),(0, 0, -5),(0, 0, 0)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(0, 4, 7),(-4, 0, -3),(-7, 3, 0)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(5),(4),(-3)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[9   sqrt(2)  -3]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(6, 0),(0, 6)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(2, 0, 0),(3, -1, 0),(-7, 3, 1)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(3, 0, 0),(0, 5, 0),(0, 0, 1/3)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(10, -15, 27),(-15, 0, sqrt(34)),(27, sqrt(34), 5/3)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(0, 0, 1),(0, 1, 0),(1, 0, 0)]`

Identify the following matrix is singular or non-singular?

`[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]`

Identify the following matrix is singular or non-singular?

`[(5, 0, 5),(1, 99, 100),(6, 99, 105)]`

Identify the following matrix is singular or non-singular?

`[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]`

Identify the following matrix is singular or non-singular?

`[(7, 5),(-4, 7)]`

Find k if the following matrix is singular:

`[(7, 3),(-2, "k")]`

Find k if the following matrix is singular:

`[(4, 3, 1),(7, "k", 1),(10, 9, 1)]`

Find k if the following matrix is singular:

`[("k" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]`

If A = `[(5, 1, -1),(3, 2, 0)]`, Find (A^T)^T.

If A = `[(7, 3, 1),(-2, -4, 1),(5, 9, 1)]`, Find (A^T)^T.

Find a, b, c if `[(1, 3/5, "a"),("b", -5, -7),(-4, "c", 0)]` is a symmetric matrix.

Find x, y, z If `[(0, -5"i", x),(y, 0, z),(3/2, -sqrt(2), 0)]` is a skew symmetric matrix.

The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:

`[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`

The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:

`[(2, 5, 1),(-5, 4, 6),(-1, -6, 3)]`

The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:

`[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]`

Construct the matrix A = [a_ij]_{3 × 3} where a_ij = i − j. State whether A is symmetric or skew-symmetric.

If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2), (0, 3)] and "C" = [(4,  3),(-1, 4),(-2, 1)]` Show that A + B = B + A

If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2), (0, 3)] and "C" = [(4,  3),(-1, 4),(-2, 1)]` Show that (A + B) + C = A + (B + C)

If A = `[(1, -2),(5, 3)], "B" = [(1, -3),(4, -7)]`, then find the matrix A – 2B + 6I, where I is the unit matrix of order 2.

If A = `[(1, 2, -3),(-3, 7, -8),(0, -6, 1)], "B" = [(9, -1, 2),(-4, 2, 5),(4, 0, -3)]` then find the matrix C such that A + B + C is a zero matrix

If A = `[(1, -2),(3, -5),(-6, 0)], "B" = [(-1, -2),(4, 2),(1, 5)] and "C" = [(2, 4),(-1, -4),(-3, 6)]`, find the matrix X such that 3A – 4B + 5X = C.

Solve the following equations for X and Y, if 3X − Y = `[(1, -1),(-1, 1)]`  and X – 3Y = `[(0, -1),(0, -1)]`.

Find matrices A and B, if `2"A" - "B" = [(6, -6, 0),(-4, 2, 1)] and "A" - 2"B" = [(3, 2, 8),(-2, 1, -7)]` 

Simplify, `costheta[(costheta, sintheta),(-sintheta, costheta)] + sintheta[(sintheta, -costheta),(costheta, sintheta)]`

If A = `[("i", 2"i"),(-3, 2)] and "B" = [(2"i", "i"),(2, -3)]`, where `sqrt(-1)` = i,, find A + B and A – B. Show that A + B is a singular. Is A – B a singular ? Justify your answer.

Find x and y, if `[(2x + y, -1, 1),(3, 4y, 4)] + [(-1, 6, 4),(3, 0, 3)] = [(3, 5, 5),(6, 18, 7)]`

If = `[(2"a" + "b", 3"a" - "b"),("c" + 2"d", 2"c" - "d")] = [(2, 3),(4, -1)]`, find a, b, c and d.

There are two book shops owned by Suresh and Ganesh. Their sales (in Rupees) for books in three subject – Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices A and B.

July sales (in Rupees), Physics Chemistry Mathematics.

A = `[(5600, 6750, 8500),(6650, 7055, 8905)]"First Row Suresh"/"Second Row Ganesh"`

August sales(in Rupees), Physics Chemistry Mathematics

B = `[(6650, 7055, 8905),(7000, 7500, 10200)]"First Row Suresh"/"Second Row Ganesh"` then,

Find the increase in sales in Rupees from July to August 2017.

There are two book shops owned by Suresh and Ganesh. Their sales (in Rupees) for books in three subject – Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices A and B.

July sales (in Rupees), Physics Chemistry Mathematics.

A = `[(5600, 6750, 8500),(6650, 7055, 8905)]"First Row Suresh"/"Second Row Ganesh"`

August sales(in Rupees), Physics Chemistry Mathematics

B = `[(6650, 7055, 8905),(7000, 7500, 10200)]"First Row Suresh"/"Second Row Ganesh"` then,

If both book shops got 10 % profit in the month of August 2017, find the profit for each book seller in each subject in that month

Evaluate: `[(3),(2),(1)][(2,-4,3)]`

Evaluate : `[2  -1   3][(4),(3),(1)]`

If A = `[(1, -3),(4, 2)], "B" = [(4, 1),(3, -2)]` show that AB ≠ BA.

If A = `[(-1, 1, 1),(2, 3, 0),(1, -3, 1)], "B" = [(2, 1, 4),(3, 0, 2),(1, 2, 1)]`. State whether AB = BA? Justify your answer.

Show that AB = BA where,

A = `[(-2, 3, -1),(-1, 2, -1),(-6, 9, -4)], "B" = [(1, 3, -1),(2, 2, -1),(3, 0, -1)]`

Show that AB = BA where, 

A = `[(costheta, - sintheta),(sintheta, costheta)], "B" = [(cosphi, -sinphi),(sinphi, cosphi)]`

If A = `[(4, 8),(-2, -4)]`, prove that A² = 0

Verify A(BC) = (AB)C of the following case:

A = `[(1, 0, 1),(2, 3, 0),(0, 4, 5)], "B" = [(2, -2),(-1, 1),(0, 3)] and "C" = [(3, 2, -1),(2, 0, -2)]`

Verify A(BC) = (AB)C of the following case:

A = `[(2, 4, 3),(-1, 3, 2)], "B" = [(2, -2),(3, 3),(-1, 1)], "C" = [(3, 1),(1, 3)]`

Verify that A(B + C) = AB + AC of the following matrix:

A = `[(4, -2),(2, 3)], "B" = [(-1, 1),(3, -2)] and "C" = [(4, 1),(2, -1)]`

Verify that A(B + C) = AB + AC of the following matrix:

A = `[(1, -1, 3),(2, 3, 2)], "B" = [(1, 0),(-2, 3),(4, 3)], "C" = [(1, 2),(-2, 0),(4, -3)]`

If A = `[(1, -2),(5, 6)], "B" = [(3, -1),(3, 7)]`, Find AB - 2I, where I is unit matrix of order 2.

If A = `[(4, 3, 2),(-1, 2, 0)], "B" = [(1, 2),(-1, 0),(1, -2)]` show that matrix AB is non singular

If A = `[(1, 2, 0),(5, 4, 2),(0, 7, -3)]`, find the product (A + I)(A − I)

A = `[(alpha, 0),(1, 1)], "B" = [(1, 0),(2, 1)]` find α, if A² = B.

If A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`, Show that A² – 4A is a scalar matrix 

If A = `[(1, 0),(-1, 7)]`, find k so that A² – 8A – kI = O, where I is a unit matrix and O is a null matrix of order 2.

If A = `[(8, 4),(10, 5)], "B" = [(5, -4),(10, -8)]` show that (A + B)² = A² + AB + B² 

If A = `[(3, 1),(-1, 2)]`, prove that A² – 5A + 7I = 0, where I is unit matrix of order 2

If A = `[(3, 4),(-4, 3)] and "B" = [(2, 1),(-1, 2)]`, show that (A + B)(A – B) = A² – B² 

If A = `[(1, 2),(-1, -2)], "B" = [(2, "a"),(-1, "b")]` and if (A + B)² = A² + B² . find values of a and b

Find matrix X such that AX = B, where A = `[(1, -2),(-2, 1)]` and B = `[(-3),(-1)]`

Find k, if A= `[(3, -2),(4, -2)]` and if A² = kA – 2I

Find x, if `[(1, "x", 1)][(1, 2, 3),(4, 5, 6),(3, 2, 5)] [(1),(-2), (3)]` = 0

Find x and y, if `{4[(2, -1, 3),(1, 0, 2)] -[(3, -3, 4),(2, 1, 1)]} [(2),(-1),(1)] = [(x),(y)]`

Find x, y, z if `{3[(2, 0),(0, 2),(2, 2)] -4[(1, 1),(-1, 2),(3, 1)]} [(1),(2)]` = `[("x" - 3),("y" - 1),(2"z")]`

If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, show that `"A"^2=[(cos2alpha, sin2alpha),(-sin2alpha, cos2alpha)]`

If A = `[(1, 2),(3, 5)]` B = `[(0, 4),(2, -1)]`, show that AB ≠ BA, but |AB| = IAl·IBI

Jay and Ram are two friends in a class. Jay wanted to buy 4 pens and 8 notebooks, Ram wanted to buy 5 pens and 12 notebooks. Both of them went to a shop. The price of a pen and a notebook which they have selected was Rs.6 and Rs.10. Using Matrix multiplication, find the amount required from each one of them

Find A^T, if A = `[(1, 3),(-4, 5)]`

Find A^T, if A = `[(2, -6, 1),(-4, 0, 5)]`

If [a_ij]_3×3 where a_ij = 2(i – j). Find A and A^T. State whether A and A^T are symmetric or skew-symmetric matrices?

If A = `[(5, -3),(4, -3),(-2, 1)]`, Prove that (2A)^T = 2A^T 

If A = `[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`, Prove that (3A)^T = 3A^T 

If A = `[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]` where i = `sqrt(-1)` Prove that A^T = – A

If A = `[(2, -3),(5, -4),(-6, 1)]`, B = `[(2, 1),(4, -1),(-3, 3)]` and C = `[(1, 2),(-1, 4),(-2, 3)]` then show that (A + B)^T = A^T + B^T

If A = `[(2, -3),(5, -4),(-6, 1)]`, B = `[(2, 1),(4, -1),(-3, 3)]` and C = `[(1, 2),(-1, 4),(-2, 3)]` then show that (A – C)^T = A^T – C^T 

If A = `[(5, 4),(-2, 3)]` and B = `[(-1, 3),(4, -1)]`, then find C^T , such that 3A – 2B + C = I, where I is the unit matrix of order 2

If A = `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find A^T + 4B^T

If A = `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find 5A^T – 5B^T

If A = `[(1, 0, 1),(3, 1, 2)]`, B = `[(2, 1, -4),(3, 5, -2)]` and C = `[(0, 2, 3),(-1, -1, 0)]`, verify that (A + 2B + 3C)^T = A^T + 2B^T + 3C^T

If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + B^T)^T = A^T + B

Prove that A + A^T is a symmetric and A – A^T is a skew symmetric matrix, where 

A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`

Prove that A + A^T is a symmetric and A – A^T is a skew symmetric matrix, where

A = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)]`

Express the following matrix as the sum of a symmetric and a skew symmetric matrix

`[(4, -2),(3, -5)]`

Express the following matrix as the sum of a symmetric and a skew symmetric matrix

`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`

If A = `[(2, -1),(3, -2),(4, 1)]` and B = `[(0, 3, -4),(2, -1, 1)]`, verify that  (AB)^T = B^T A^T 

If A = `[(2, -1),(3, -2),(4, 1)]` and B = `[(0, 3, -4),(2, -1, 1)]`, verify that (BA)^T = A^T B^T

If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, show that A^TA = I, where I is the unit matrix of order 2

Select the correct option from the given alternatives:

Given A = `[(1, 3),(2, 2)]`, I = `[(1, 0),(0, 1)]` if A – λI is a singular matrix then _______

Select the correct option from the given alternatives:

Consider the matrices A = `[(4, 6, -1),(3, 0, 2),(1, -2, 5)]`, B = `[(2, 4),(0, 1),(-1, 2)]`, C = `[(3),(1),(2)]` out of the given matrix product ________

i) (AB)^TC
ii) C^TC(AB)^T  
iii) C^TAB
iv) A^TABB^TC

Select the correct option from the given alternatives:

If A and B are square matrices of equal order, then which one is correct among the following?

Select the correct option from the given alternatives:

If A = `[(1, 2, 2),(2, 1, -2),("a", 2, "b")]` is a matrix satisfying the equation AA^T = 9I, where I is the identity matrix of order 3, then the ordered pair (a, b) is equal to ________

Select the correct option from the given alternatives:

If A = `[(alpha, 2),(2, alpha)]` and |A³| = 125, then α = _______

Select the correct option from the given alternatives:

If `[(5, 7),(x, 1),(2, 6)] - [(1, 2),(-3, 5),(2, y)] = [(4, 5),(4, -4),(0, 4)]` then __________

Select the correct option from the given alternatives:

If A + B = `[(7, 4),(8, 9)]` and A − B = `[(1, 2),(0, 3)]` then the value of A is _______

Select the correct option from the given alternatives:

If `[("x", 3"x" - "y"),("zx" + "z", 3"y" - "w")] = [(3, 2),(4, 7)]` then ______

Select the correct option from the given alternatives:

For suitable matrices A, B, the false statement is _____

Select the correct option from the given alternatives:

If A = `[(-2, 1),(0, 3)]` and f(x) = 2x² – 3x, then f(A) = ………

Answer the following question:

If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find B + C – A

Answer the following question:

If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find 2A + B – 5C

Answer the following question:

If f(α) = A = `[(cosalpha, -sinalpha, 0),(sinalpha, cosalpha, 0),(0, 0, 1)]`, Find f(– α)

Answer the following question:

If f (α) = A = `[(cosalpha, -sinalpha, 0),(sinalpha, cosalpha, 0),(0, 0, 1)]`, Find f(–α) + f(α)

Answer the following question:

Find matrices A and B, where 2A – B = `[(1, -1),(0, 1)]` and A + 3B = `[(1, -1),(0, 1)]`

Answer the following question:

Find matrices A and B, where 3A – B = `[(-1, 2, 1),(1, 0, 5)]` and A + 5B = `[(0, 0, 1),(-1, 0, 0)]`

Answer the following question:

If A = `[(2, -3),(3, -2),(-1, 4)]`, B = `[(-3, 4, 1),(2, -1, -3)]` Verify (A + 2B^T)^T = A^T + 2B

Answer the following question:

If A = `[(2, -3),(3, -2),(-1, 4)]`, B = `[(-3, 4, 1),(2, -1, -3)]` Verify (3A - 5B^T)^T = 3A^T – 5B

Answer the following question:

If A = `[(cosalpha, -sinalpha),(sinalpha, cosalpha)]` and A + A^T = I, where I is unit matrix 2 × 2, then find the value of α

Answer the following question:

If A = `[(1, 2),(3, 2),(-1, 0)]` and B = `[(1, 3, 2),(4, -1, -3)]`, show that AB is singular.

Answer the following question:

If A = `[(1, 2, 3),(2, 4, 6),(1, 2, 3)]`, B = `[(1, -1, 1),(-3, 2, -1),(-2, 1, 0)]`, show that AB and BA are both singular matrices

Answer the following question:

If A = `[(1, -1, 0),(2, 3, 4),(0, 1, 2)]`, B = `[(2, 2, -4),(-4, 2, -4),(2, -1, 5)]`, show that BA = 6I

Answer the following question:

If A = `[(2, 1),(0, 3)]`, B = `[(1, 2),(3, -2)]`, verify that |AB| = |A||B|

Answer the following question:

If A_α = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, show that A_α . A_β = A_α+β 

Answer the following question:

If A = `[(1, omega),(omega^2, 1)]`, B = `[(omega^2, 1),(1, omega)]`, where ω is a complex cube root of unity, then show that AB + BA + A −2B is a null matrix

Answer the following question:

If A = `[(2, -2, -4),(-1, 3, 4),(1, -2, -3)]` show that A² = A

Answer the following question:

If A = `[(4, -1, -4),(3, 0, -4),(3, -1, -3)]`, show that A² = I

Answer the following question:

If A = `[(3, -5),(-4, 2)]`, show that A² – 5A – 14I = 0

Answer the following question:

If A = `[(2, -1),(-1, 2)]`, show that A² – 4A + 3I = 0

Answer the following question:

if A = `[(-3, 2),(2, -4)]`, B = `[(1, x),(y, 0)]`, and (A + B)(A – B) = A² – B² , find x and y

Answer the following question:

If A = `[(0, 1),(1, 0)]` and B = `[(0, -1),(1, 0)]` show that (A + B)(A – B) ≠ A² – B² 

Answer the following question:

If A = `[(2, -1),(3, -2)]`, find A³  

Answer the following question:

Find x, y if, `[(0, -1, 4)]{2[(4, 5),(3, 6),(2, -1)] + 3[(4, 3),(1, 4),(0, -1)]} = [(x, y)]`

Answer the following question:

Find x, y if, `{-1 [(1, 2, 1),(2, 0, 3)] + 3[(2, -3, 7),(1, -1, 3)]} [(5),(0),(-1)] = [(x),(y)]`

Answer the following question:

Find x, y, z if `{5[(0, 1),(1, 0),(1, 1)] -3[(2, 1),(3, -2),(1, 3)]} [(2),(1)] = [(x - 1),(y + 1),(2z)]`

Answer the following question:

Find x, y, z if `{[(1, 3, 2),(2, 0, 1),(3, 1, 2)] + 2[(3, 0, 2),(1, 4, 5),(2, 1, 0)]} [(1),(2),(3)] = [(x),(y),(z)]`

Answer the following question:

If A = `[(2, 1, -3),(0, 2, 6)]`, B = `[(1, 0, -2),(3, -1, 4)]`, find AB^T and A^TB

Answer the following question:

If A = `[(2, -4),(3, -2),(0, 1)]`, B = `[(1, -1, 2),(-2, 1, 0)]`, show that (AB)^T = B^TA^T

Answer the following question:

If A = `[(3, -4),(1, -1)]`, prove that Aⁿ = `[(1 + 2"n", -4"n"),("n", 1 - 2"n")]`, for all n ∈ N

Answer the following question:

If A = `[(costheta, sintheta),(-sintheta, costheta)]`, prove that Aⁿ = `[(cos"n"theta, sin"n"theta),(-sin"n"theta, cos"n"theta)]`, for all n ∈ N

Answer the following question:

Two farmers Shantaram and Kantaram cultivate three crops rice,wheat and groundnut. The sale (In Rupees) of these crops by both the farmers for the month of April and may 2008 is given below,

Find The total sale in rupees for two months of each farmer for each crop

Answer the following question:

Two farmers Shantaram and Kantaram cultivate three crops rice,wheat and groundnut. The sale (In Rupees) of these crops by both the farmers for the month of April and may 2008 is given below,

Find the increase in sales from April to May for every crop of each farmer.