NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 3 - Matrices [Latest edition]

Construct a matrix A = [a_ij]_2×2 whose elements a_ij are given by a_ij = e^2ix sin jx.

If A = `[(2, 3),(1, 2)]`, B = `[(1, 3, 2),(4, 3, 1)]`, C = `[(1),(2)]`, D = `[(4, 6, 8),(5, 7, 9)]`, then which of the sums A + B, B + C, C + D and B + D is defined?

Show that a matrix which is both symmetric and skew symmetric is a zero matrix.

If `[(2x, 3)] [(1, 2),(-3, 0)] [(x),(8)]` = 0, find the valof x.

If A is 3 × 3 invertible matrix, then show that for any scalar k (non-zero), kA is invertible and `("kA")^-1 = 1/"k" "A"^-1`

Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`

If A = `[(1, 3, 2),(2, 0, -1),(1, 2, 3)]`, then show that A satisfies the equation A³ – 4A² – 3A + 11I = O.

Let A = `[(2, 3),(-1, 2)]`. Then show that A² – 4A + 7I = O. Using this result calculate A⁵ also.

If A and B are square matrices of the same order, then (A + B)(A – B) is equal to ______.

If A = `[(2, -1, 3),(-4, 5, 1)]` and B = `[(2, 3),(4, -2),(1, 5)]`, then ______.

The matrix A = `[(0, 0, 5),(0, 5, 0),(5, 0, 0)]` is a ______.

If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.

If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.

If A and B are matrices of same order, then (3A –2B)′ is equal to______.

Addition of matrices is defined if order of the matrices is ______.

If two matrices A and B are of the same order, then 2A + B = B + 2A.

Matrix subtraction is associative

For the non singular matrix A, (A′)^–1 = (A^–1)′.

AB = AC ⇒ B = C for any three matrices of same order.

If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

In the matrix A = `[("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)]`, write: The order of the matrix A

In the matrix A = `[("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)]`, write: The number of elements

In the matrix A = `[("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)]`, write: elements a₂₃, a₃₁, a₁₂ 

Construct a_{2 × 2} matrix where a_ij = `("i" - 2"j")^2/2`

Construct a_{2 × 2} matrix where a_ij = |–2i + 3j|

Construct a 3 × 2 matrix whose elements are given by a_ij = e^i.x sinjx.

Find the values of a and b if A = B, where A = `[("a" + 4, 3"b"),(8, -6)]`, B = `[(2"a" + 2, "b"^2 + 2),(8, "b"^2 - 5"b")]`

If possible, find the sum of the matrices A and B, where A = `[(sqrt(3), 1),(2, 3)]`, and B = `[(x, y, z),(a, "b", 6)]`

If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find X + Y

If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find 2X – 3Y

If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find A matrix Z such that X + Y + Z is a zero matrix

Find non-zero values of x satisfying the matrix equation:

`x[(2x, 2),(3, x)] + 2[(8, 5x),(4, 4x)] = 2[(x^2 + 8, 24),(10, 6x)]`

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

Find the value of x if `[(1, x, 1)] [(1, 3, 2),(2, 5,1),(15, 3, 2)] [(1),(2),(x)]` = 0

Show that A = `[(5, 3),(-1, -2)]` satisfies the equation A² – 3A – 7I = O and hence find A^–1.

Find the matrix A satisfying the matrix equation:

`[(2, 1),(3, 2)] "A" [(-3, 2),(5, -3)] = [(1, 0),(0, 1)]`

Find A, if `[(4),(1),(3)]` A = `[(-4, 8,4),(-1, 2, 1),(-3, 6, 3)]`

If A = `[(3, -4),(1, 1),(2, 0)]` and B = `[(2, 1, 2),(1, 2, 4)]`, then verify (BA)² ≠ B²A² 

If possible, find BA and AB, where A = `[(2, 1, 2),(1, 2, 4)]`, B = `[(4, 1),(2, 3),(1, 2)]`

Show by an example that for A ≠ O, B ≠ O, AB = O

Given A = `[(2, 4, 0),(3, 9, 6)]` and B = `[(1, 4),(2, 8),(1, 3)]` is (AB)′ = B′A′? 

Solve for x and y: `x[(2),(1)] + y[(3),(5)] + [(-8),(-11)]` = O

If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.

2X + 3Y = `[(2, 3),(4, 0)]`, 3Y + 2Y = `[(-2, 2),(1, -5)]`

If A = `[(3, 5)]`, B = `[(7, 3)]`, then find a non-zero matrix C such that AC = BC.

Give an example of matrices A, B and C such that AB = AC, where A is nonzero matrix, but B ≠ C.

If A = `[(1, 2),(-2, 1)]`, B = `[(2, 3),(3, -4)]` and C = `[(1, 0),(-1, 0)]`, verify: (AB)C = A(BC)

If A = `[(1, 2),(-2, 1)]`, B = `[(2, 3),(3, -4)]` and C = `[(1, 0),(-1, 0)]`, verify: A(B + C) = AB + AC

If P = `[(x, 0, 0),(0, y, 0),(0, 0, z)]` and Q = `[("a", 0, 0),(0, "b", 0),(0, 0, "c")]`, prove that PQ = `[(x"a", 0, 0),(0, y"b", 0),(0, 0, z"c")]` = QP

If: `[(2, 1, 3)] [(-1, 0, -1),(-1, 1, 0),(0, 1, 1)] [(1),(0),(-1)]` = A, find A

If A = `[(2, 1)]`, B = `[(5, 3, 4),(8, 7, 6)]` and C = `[(-1, 2, 1),(1, 0, 2)]`, verify that A(B + C) = (AB + AC).

If A = `[(1, 0, -1),(2, 1, 3 ),(0, 1, 1)]`, then verify that A² + A = A(A + I), where I is 3 × 3 unit matrix.

If A = `[(0, -1, 2),(4, 3, -4)]` and B = `[(4, 0),(1, 3),(2, 6)]`, then verify that: (A′)′ = A

If A = `[(0, -1, 2),(4, 3, -4)]` and B = `[(4, 0),(1, 3),(2, 6)]`, then verify that: (A′)′ = (AB)' = B'A'

If A = `[(0, -1, 2),(4, 3, -4)]` and B = `[(4, 0),(1, 3),(2, 6)]`, then verify that: (kA)' = (kA')

If A = `[(1, 2),(4, 1),(5, 6)]` B = `[(1, 2),(6, 4),(7, 3)]`, then verify that: (2A + B)′ = 2A′ + B′

If A = `[(1, 2),(4, 1),(5, 6)]` B = `[(1, 2),(6, 4),(7, 3)]`, then verify that: (A – B)′ = A′ – B′

Show that A′A and AA′ are both symmetric matrices for any matrix A.

Let A and B be square matrices of the order 3 × 3. Is (AB)² = A²B²? Give reasons.

Show that if A and B are square matrices such that AB = BA, then (A + B)² = A² + 2AB + B².

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: A + (B + C) = (A + B) + C

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: A(BC) = (AB)C

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: (a + b)B = aB + bB

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: a(C – A) = aC – aA

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: (A^T)^T = A

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: (bA)^T = bA^T

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: (AB)^T = B^TA^T

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: (A – B)C = AC – BC 

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: (A – B)^T = A^T – B^T 

If A = `[(costheta, sintheta),(-sintheta, costheta)]`, then show that A² = `[(cos2theta, sin2theta),(-sin2theta, cos2theta)]`

If A = `[(0, -x),(x, 0)]`, B = `[(0, 1),(1, 0)]` and x² = –1, then show that (A + B)² = A² + B². 

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

Prove by Mathematical Induction that (A′)ⁿ = (Aⁿ)′, where n ∈ N for any square matrix A.

Find inverse, by elementary row operations (if possible), of the following matrices

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

Find inverse, by elementary row operations (if possible), of the following matrices

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

If `[(xy, 4),(z + 6, x + y)] = [(8, w),(0, 6)]`, then find values of x, y, z and w.

If A = `[(1, 5),(7, 12)]` and B  `[(9, 1),(7, 8)]`, find a matrix C such that 3A + 5B + 2C is a null matrix.

If A = `[(3, -5),(-4, 2)]`, then find A² – 5A – 14I. Hence, obtain A³.

Find the values of a, b, c and d, if `3[("a", "b"),("c", "d")] = [("a", 6),(-1, 2"d")] + [(4, "a" + "b"),("c" + "d", 3)]`

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

If A = `[(1, 2),(4, 1)]`, find A² + 2A + 7I.

If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A^–1 = A′, find value of α

If the matrix `[(0, "a", 3),(2, "b", -1),("c", 1, 0)]`, is a skew symmetric matrix, find the values of a, b and c.

If P(x) = `[(cosx, sinx),(-sinx, cosx)]`, then show that P(x) . (y) = P(x + y) = P(y) . P(x)

If A is square matrix such that A² = A, show that (I + A)³ = 7A + I..

If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.

If AB = BA for any two square matrices, prove by mathematical induction that (AB)ⁿ = AⁿBⁿ 

Find x, y, z if A = `[(0, 2y, z),(x, y, -z),(x, -y, z)]` satisfies A′ = A^–1.

If possible, using elementary row transformations, find the inverse of the following matrices

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

If possible, using elementary row transformations, find the inverse of the following matrices

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

If possible, using elementary row transformations, find the inverse of the following matrices

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

Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.

The matrix P = `[(0, 0, 4),(0, 4, 0),(4, 0, 0)]`is a ______.

Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is ______.

If `[(2x + y, 4x),(5x - 7, 4x)] = [(7, 7y - 13),(y, x + 6)]`, then the value of x + y is ______.

If A = `1/pi [(sin^-1(xpi), tan^-1(x/pi)),(sin^-1(x/pi), cot^-1(pix))]`, B = `1/pi [(-cos^-1(x/pi), tan^-1 (x/pi)),(sin^-1(x/pi),-tan^-1(pix))]`, then A – B is equal to ______.

If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the order of matrix (5A – 2B) is ______.

If A = `[(0, 1),(1, 0)]`, then A² is equal to ______.

If matrix A = [a_ij]_2×2, where a_ij `{:(= 1  "if i" ≠ "j"),(= 0  "if i" = "j"):}` then A² is equal to ______.

The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.

The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.

If A is matrix of order m × n and B is a matrix such that AB′ and B′A are both defined, then order of matrix B is ______.

If A and B are matrices of same order, then (AB′ – BA′) is a ______.

If A is a square matrix such that A² = I, then (A – I)³ + (A + I)³ –7A is equal to ______.

For any two matrices A and B, we have ______.

On using elementary column operations C₂ → C₂ – 2C₁ in the following matrix equation `[(1, -3),(2, 4)] = [(1, -1),(0, 1)] [(3, 1),(2, 4)]`, we have: ______.

On using elementary row operation R₁ → R₁ – 3R₂ in the following matrix equation: `[(4, 2),(3, 3)] = [(1, 2),(0, 3)] [(2, 0),(1, 1)]`, we have: ______.

______ matrix is both symmetric and skew-symmetric matrix.

Sum of two skew-symmetric matrices is always ______ matrix.

The negative of a matrix is obtained by multiplying it by ______.

The product of any matrix by the scalar ______ is the null matrix.

A matrix which is not a square matrix is called a ______ matrix.

Matrix multiplication is ______ over addition.

If A is a symmetric matrix, then A³ is a ______  matrix.

If A is a skew-symmetric matrix, then A² is a ______.

If A and B are square matrices of the same order, then (AB)′ = ______.

If A and B are square matrices of the same order, then (kA)′ = ______. (k is any scalar)

If A and B are square matrices of the same order, then [k (A – B)]′ = ______.

If A is skew-symmetric, then kA is a ______. (k is any scalar)

If A and B are symmetric matrices, then AB – BA is a ______.

If A and B are symmetric matrices, then BA – 2AB is a ______.

If A is symmetric matrix, then B′AB is ______.

If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.

In applying one or more row operations while finding A^–1 by elementary row operations, we obtain all zeros in one or more, then A^–1 ______.

A matrix denotes a number.

Matrices of any order can be added.

Two matrices are equal if they have same number of rows and same number of columns.

Matrices of different orders can not be subtracted.

Matrix addition is associative as well as commutative.

Matrix multiplication is commutative.

A square matrix where every element is unity is called an identity matrix.

If A and B are two square matrices of the same order, then A + B = B + A.

If A and B are two matrices of the same order, then A – B = B – A.

If matrix AB = O, then A = O or B = O or both A and B are null matrices.

Transpose of a column matrix is a column matrix.

If A and B are two square matrices of the same order, then AB = BA.

If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.

If A and B are any two matrices of the same order, then (AB)′ = A′B′.

If (AB)′ = B′ A′, where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.

If A, B and C are square matrices of same order, then AB = AC always implies that B = C

AA′ is always a symmetric matrix for any matrix A.

If A = `[(2, 3, -1),(1, 4, 2)]` and B = `[(2, 3),(4, 5),(2, 1)]`, then AB and BA are defined and equal.

If A is skew-symmetric matrix, then A² is a symmetric matrix.

(AB)^–1 = A^–1. B^–1, where A and B are invertible matrices satisfying commutative property with respect to multiplication.