Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 11 TN Board chapter 7 - Matrices and Determinants [Latest edition]

Construct an m × n matrix A = [a_ij], where a_ij is given by

a_ij = `("i" - 2"j")^2/2` with m = 2, n = 3

Construct an m × n matrix A = [a_ij], where a_ij is given by

a_ij = `|3"i" - 4"j"|/4` with m = 3, n = 4

Find the values of p, q, r, and s if

`[("p"^2 - 1, 0, - 31 - "q"^3),(7, "r" + 1, 9),(- 2, 8, "s" - 1)] = [(1, 0, -4),(7, 3/2, 9),(-2, 8, -pi)]`

Determine the value of x + y if `[(2x + y, 4x),(5x - 7, 4x)] = [(7, 7y - 13),(y, x + 6)]`

Determine the matrices A and B if they satisfy 2A – B + `[(6, - 6, 0),(- 4, 2, 1)]` = 0 and A – 2B = `[(3, 2, 8),(-2, 1, -7)]`

If A = `[(1, "a"),(0, 1)]`, then compute A⁴ 

Consider the matrix A_α = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Show that `"A"_alpha "A"_beta = "A"_((alpha + beta))`

Consider the matrix A_α = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Find all possible real values of α satisfying the condition `"A"_alpha + "A"_alpha^"T"` = I

If A = `[(4, 2),(-1, x)]` and such that (A – 2I)(A – 3I) = 0, find the value of x

If A = `[(1, 0, 0),(0, 1, 0),("a", "b", - 1)]`, show that A² is a unit matrix

If A = `[(1, 0, 2), (0, 2, 1), (2, 0, 3)]` and A³ – 6A² + 7A + kI = 0, find the value of k

Give your own examples of matrices satisfying the following conditions:
A and B such that AB ≠ BA

Give your own examples of matrices satisfying the following conditions:
A and B such that AB = 0 = BA, A ≠ 0 and B ≠ 0

Give your own examples of matrices satisfying the following conditions:
A and B such that AB = 0 and BA ≠ 0

Show that f(x) f(y) = f(x + y), where f(x) = `[(cosx, -sinx, 0),(sinx, cosx, 0),(0, 0, 1)]`

If A is a square matrix such that A² = A, find the value of 7A – (I + A)³

Verify the property A(B + C) = AB + AC, when the matrices A, B, and C are given by A = `[(2, 0, -3),(1, 4, 5)]`, B = `[(3, 1),(-1, 0),(4, 2)]` and C = `[(4, 7),(2, 1),(1,-1)]`

Find the matrix A which satisfies the matrix relation `"A"= [(1, 2, 3),(4, 5, 6)] = [(-7, -8, -9),(2, 4, 6)]`

If A^T = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`, veriy the following 

(A + B)^T = A^T + B^T = B^T + A^T

If A^T = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`, veriy the following 

(A – B)^T = A^T – B^T

If A^T = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`, veriy the following 

(B^T)^T = B

If A is a 3 × 4 matrix and B is a matrix such that both A^TB and BA^T are defined, what is the order of the matrix B?

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:

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

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:

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

Find the matrix A such that `[(2, -1),(1, 0),(-3, 4)]"A"^"T" = [(-1, -8, -10),(1, 2, -5),(9, 22, 15)]`

If A = `[(1, 2, 2),(2, 1, -2),(x, 2, y)]` is a matrix such that AA^T = 9I, find the values of x and y

For what value of x, the matrix A = `[(0, 1, -2),(-1, 0, x^3),(2, -3, 0)]` is skew – symmetric

If `[(0, "p", 3),(2, "q"^2, -1),("r", 1, 0)]` is skew – symmetric find the values of p, q and r

Construct the matrix A = [a_ij]_3×3, where a_ij = 1 – j. State whether A is symmetric or skew–symmetric

Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix

If A and B are symmetric matrices of same order, prove that AB + BA is a symmetric matrix

If A and B are symmetric matrices of same order, prove that AB – BA is a skew-symmetric matrix

A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins and almonds. Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds. Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds. Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds. The cost of 50 gm of cashew nuts is ₹ 50, 50 gm of raisins is ₹ 10, and 50 gm of almonds is ₹ 60. What is the cost of each gift pack?

Without expanding the determinant, prove that `|("s", "a"^2, "b"^2 + "c"^2),("s", "b"^2, "c"^2 + "a"^2),("s", "c"^2, "a"^2 + "b"^2)|` = 0

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

Prove that `|("a"^2, "bc", "ac" + "c"^2),("a"^2 + "ab", "b"^2, "ac"),("ab", "b"^2 + "bc", "c"^2)| = 4"a"^2"b"^2"c"^2`

Prove that `|(1 + "a", 1, 1),(1, 1 + "b", 1),(1, 1, 1 + "c")| = "abc"(1 + 1/"a" + 1/"b" + 1/"c")`

Prove that `|(sec^2theta, tan^2theta, 1),(tan^2theta, sec^2theta, -1),(38, 36, 2)|` = 0

Show that `|(x + 2"a", y + 2"b", z + 2"c"),(x, y, z),("a", "b", "c")|` = 0

Write the general form of a 3 × 3 skew-symmetric matrix and prove that its determinant is 0

If `|("a", "b", "a"alpha + "b"),("b", "c", "b"alpha + "c"),("a"alpha + "b", "b"alpha + "c", 0)|` = 0, prove that a, b, c are in G. P or α is a root of ax² + 2bx + c = 0

Prove that `|(1, "a", "a"^2 - "bc"),(1, "b", "b"^2 - "ca"),(1, "c", "c"^2 - "ab")|` = 0

If a, b, c are p^th, q^th and r^th terms of an A.P, find the value of `|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|`

Show that `|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|` is divisiible by x⁴ 

If a, b, c, are all positive, and are p^th, q^th and r^th terms of a G.P., show that `|(log"a", "p", 1),(log"b", "q", 1),(log"c", "r", 1)|` = 0

Find the value of `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` if x, y, z ≠ 1

If A = `[(1/2, alpha),(0, 1/2)]`, prove that `sum_("k" = 1)^"n" det("A"^"k") = 1/3(1 - 1/4)` 

Without expanding, evaluate the following determinants:

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

Without expanding, evaluate the following determinants:

`|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|`

If A is a Square, matrix, and |A| = 2, find the value of |A A^T|

If A and B are square matrices of order 3 such that |A| = –1 and |B| = 3, find the value of |3AB|

If λ = – 2, determine the value of `|(0, lambda, 1),(lambda^2, 0, 3lambda^2 + 1),(-1, 6lambda - 1, 0)|`

Determine the roots of the equation `|(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)|` = 0

Verify that det(AB) = (det A)(det B) for A = `[(4, 3, -2),(1, 0, 7),(2, 3, -5)]` and B = `[(1, 3, 3),(-2, 4, 0),(9, 7, 5)]`

Using cofactors of elements of second row, evaluate |A|, where A = `[(5, 3, 8),(2, 0, 1),(1, 2, 3)]`

Solve the following problems by using Factor Theorem:

Show that `|(x, "a", "a"),("a", x, "a"),("a", "a", x)|` = (x – a)² (x + 2a)

Show that `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|` = 8abc

Solve that `|(x + "a", "b", "c"),("a", x + "b", "c"),("a", "b", x + "c")|` = 0

Show that `|("b" + "C", "a", "a"^2),("c" + "a", "b", "b"^2),("a" + "b", "c", "c"^2)|` = (a + b + c)(a – b)(b – c)(c – a)

Solve `|(4 - x, 4 + x, 4 +  x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)|` = 0

Show that `|(1, 1, 1),(x, y, z),(x^2, y^2, z^2)|` = (x – y)(y – z)(z – x)

Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3)

If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k

Identify the singular and non-singular matrices:

`[(1, 2, 3),(4, 5, 6),(7, 8, 9)]`

Identify the singular and non-singular matrices:

`[(2, -3, 5),(6, 0, 4),(1, 5, -7)]`

Identify the singular and non-singular matrices:

`[(0, "a" - "b", "k"),("b" - "a", 0, 5),(-"k", -5, 0)]`

Determine the values of a and b so that the following matrices are singular:

A = `[(7, 3),(-2, "a")]`

Determine the values of a and b so that the following matrices are singular:

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

If cos 2θ = 0, determine  `[(theta, costheta, sintheta),(costheta, sintheta, 0),(sintheta, 0, costheta)]^2`

Find the value of the product: `|(log_3 64, log_4 3),(log_3 8, log_4 9)| xx |(log_2 3, log_8 3),(log_3 4, log_3 4)|`

Choose the correct alternative:
If a_ij =  (3i – 2j) and A = [a_ij]_{3 × 2} is

Choose the correct alternative:
What must be the matrix X, if `2"X" + [(1, 2),(3, 4)] = [(3, 8),(7, 2)]`?

Choose the correct alternative:
Which one of the following is not true about the matrix `[(1, 0, 0),(0, 0, 0),(0, 0, 5)]`?

Choose the correct alternative:
If A and B are two matrices such that A + B and AB are both defined, then

Choose the correct alternative:
if A = `[(lambda, 1),(-1, -lambda)]`, then for what value of λ, A² = 0 ?

Choose the correct alternative:
If A = `[(1, -1),(2, -1)]`, B = `[("a", 1),("b", -1)]` and (A + B)² = A² + B², then the values of a and b are

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

Choose the correct alternative:
If A is a square matrix, then which of the following is not symmetric?

Choose the correct alternative:
If A and B are symmetric matrices of order n, where (A ≠ B), then

Choose the correct alternative:
If A = `[("a", x),(y, "a")]` and if xy = 1, then det(AA^T) is equal to

Choose the correct alternative:
The value of x, for which the matrix A = `[("e"^(x - 2), "e"^(7 + x)),("e"^(2 + x), "e"^(2x + 3))]` is singular

Choose the correct alternative:
If the points (x, – 2), (5, 2), (8, 8) are collinear, then x is equal to

Choose the correct alternative:
If `|(2"a", x_1, y_1),(2"b", x_2, y_2),(2"c", x_3, y_3)| = "abc"/2 ≠ 0`, then the area of the triangle whose vertices are `(x_1/"a", y_1/"a"), (x_2/"b", y_2/"b"), (x_3/"c", y_3/"c")` is

Choose the correct alternative:
If the square of the matrix `[(alpha, beta),(γ, - alpha)]` is the unit matrix of order 2, then α, β, and γ should

Choose the correct alternative:
if Δ = `|("a", "b", "c"),(x, y, z),("p", "q", "r")|` then  `|("ka", "kb","kc"),("k"x, "k"y, "k"z),("kp", "kq", "kr")|` is

Choose the correct alternative:
A root of the equation `|(3 - x, -6, 3),(-6, 3 - x, 3),(3, 3, -6 - x)|` = 0 is

Choose the correct alternative:
The value of the determinant of A = `[(0, "a", -"b"),(-"a", 0, "c"),("b", -"c", 0)]` is

Choose the correct alternative:
If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in geometric progression with the same common ratio, then the points (x₁, y₁), (x₂, y₂), (x₃, y₃) are

Choose the correct alternative:
If ⌊.⌋ denotes the greatest integer less than or equal to the real number under consideration and – 1 ≤ x < 0, 0 ≤ y < 1, 1 ≤ z ≤ 2, then the value of the determinant `[([x] + 1, [y], [z]),([x], [y] + 1, [z]),([x], [y], [z] + 1)]`

Choose the correct alternative:
If a ≠ b, b, c satisfy `|("a", 2"b", 2"c"),(3, "b", "c"),(4, "a", "b")|` = 0, then abc =

Choose the correct alternative:
If A = `|(-1, 2, 4),(3, 1, 0),(-2, 4, 2)|` and B = `|(-2, 4, 2),(6, 2, 0),(-2, 4, 8)|`, then B is given by

Choose the correct alternative:
If A is skew-symmetric of order n and C is a column matrix of order n × 1, then C^T AC is

Choose the correct alternative:
The matrix A satisfying the equation `[(1, 3),(0, 1)] "A" = [(1, 1),(0, -1)]` is

Choose the correct alternative:
If A + I = `[(3, -2),(4, 1)]`, then (A + I)(A – I) is equal to

Choose the correct alternative:
Let A and B be two symmetric matrices of same order. Then which one of the following statement is not true?