Nootan solutions for Mathematics [English] Class 10 ICSE chapter 8 - Matrices [null edition]

If a matrix has 10 elements, what are the possible orders it can have?

If a matrix has 24 elements, what are the possible orders it can have?

Construct a 2 × 2 matrix whose elements in the ith row and jth column are given by:

`a_(ij) = (3i - j)/2`

Construct a 2 × 2 matrix whose elements in the ith row and jth column are given by:

a_ij = i + j

Construct a 2 x 2 matrix whose elements a_ij are given by `((i + 2j)^2)/(2)`.

Construct a 2 × 2 matrix whose elements in the ith row and jth column are given by:

a_ij = i × j

Construct a 2 × 2 matrix whose elements in the ith row and jth column are given by:

a_ij = (−3i + j)

Construct a 2 × 2 matrix whose elements in the ith row and jth column are given by:

a_ij = `(3i - 2j)/2`

Find x, y, z and t where, `[(z + t, x - y),(z - t, x + y)] = [(5, 3),(1, -1)]`

Find the values of ‘x’, ‘y’, ‘z’ and ‘a’ which satisfy the following matrix equations:

`[(x + 3, 2y + x),(z - 1, 4a - 6)] = [(0, -7),(3, 2a)]`

Find the values of ‘x’, ‘y’, ‘z’ and ‘a’ which satisfy the following matrix equations:

`[(x, 3x - y),(2x + z, 3y - a)] = [(3, 2),(4, 7)]`

A = `[(5, a^3),(a^2, 1)], "B" = [(5, -27),(9, 1)]`. For what values of ‘a’ does equality occur?

If `[(x + y + z),(x + y),(y + z)] = [(7),(5),(3)]`, find the values of x, y and z.

Find the values of x, y, z if `[(2x + y, x - y),(x - z, x + y + z)] = [(10, -1),(2, 8)]`.

Find the values of x and y if `[(3, 4 + x),(7 + y, 0)] = [(3, 2),(6, 0)]`.

Find x, y, z, and w if `[(x - y, 2z + w),(2x - y, 2x + w)] = [(5, 3),(12, 15)]`.

Simplify:

`[(-3, -2),(2, 0)] + [(3, 2),(-2, 0)]`

Simplify:

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

Find x if, `[(4, 5),(-3, 6)] + x = [(10, -2),(1, 4)]`

Prove that (A + B) + C= A + (B + C), when A = `[(0, -1, 2),(3, 4, -5)], B = [(-2, 0, 3),(4, -5, 6)] and C = [(4, 7, -2),(0, -5, 1)]`

Let A = `[(2, 4),(3, 2)], B = [(1, 3),(-2, 5)]`, find: A + B

 Let A = `[(2, 4),(3, 2)], B = [(1, 3),(-2, 5)]`, find: A − B

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

Find the following:

A + B

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

Find the following:

A − B

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

Find the following:

(A + B) + С

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

Find the following:

B + C

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

Find the following:

A + (B + C)

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

Find the following:

(A − B) − С

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

Find the following:

A − (B − С)

If A = `[(2, 1),(-1, 3)], B = [(-3, 2),(4, 1)], C = [(-3, 0),(4, 1)]`, verify the following:

A + B = B + A

If A = `[(2, 1),(-1, 3)], B = [(-3, 2),(4, 1)], C = [(-3, 0),(4, 1)]`, verify the following:

(A + B) + C = A + (B + C)

If A = `[(1, 2, 2),(-3, -1, 0)], B = [(1, 0, 1),(2, 1, 3)]`, find the matrix C such that A + B + C is a zero matrix.

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

Solve for x and y, if `2((1, 3),(0, x)) + ((y, 0),(1, 2)) = ((5, 6),(1, 8))`.

If `x[(2), (3)] + y[(-1),(1)] = [(10), (5)]`, find values of x and y.

Solve the matrix equation `((x^2),(y^2)) - 3((x),(2y)) = ((-2),(-9))`.

If A = `((1, 3),(2, 1),(3, -1))` and B = `((2, 1),(1, 2),(1, 1)),` then find the matrix C such that A + B + C is a zero matrix.

Find the values of x and y from the following equation:

`((x - y, 2, -2),(4, x, 6)) + ((3, -2, 2),(1, 0, -1)) = ((6, 0, 0),(5, 2x + y, 5))`

Compute the product:

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

If A = `[(3, 1),(-1, 2)], B = [(1, 0),(0, 2)]`, find AB.

 If A = `[(3, 1),(-1, 2)], B = [(1, 0),(0, 2)]`, find BA.

 If A = `[(3, 1),(-1, 2)], B = [(1, 0),(0, 2)]`, find A².

If A = `[(3, 1),(-1, 2)], B = [(1, 0),(0, 2)],` find B².

If A = `[(1, -2),(3, 0)], B = [(0, -2),(2, 1)]` find AB.

If A = `[(1, -2),(3, 0)], B = [(0, -2),(2, 1)]` find BA.

If A = `[(1, -2),(3, 0)], B = [(0, -2),(2, 1)]` is AB = BA? Also, conclude your result.

If A = `[(1, 0),(2, 1)], B = [(2, 3),(-1, 0)]`, find A² + B² + AB.

Evaluate:

`[(4 sin 30°, 2 cos 60°),(sin 90°, 2 cos 0°)] [(4, 5),(5, 4)]`

If A = `[(1, -2),(2, -1)], B = [(3, 2),(-2, 1)], C = [(4, 5),(5, 4)]`, compute A(B + C).

If A = `[(1, -2),(2, -1)], B = [(3, 2),(-2, 1)], C = [(4, 5),(5, 4)]`, compute (B + C)A.

If A = `[(3, 2),(7, 4)], B = [(0, 5),(2, 3)]`, find AB + ВА.

If A = `[(2, 1),(0, -2)], B = [(4, 1),(-3, -2)], C = [(-3, 2),(-1, 4)]`, find A² − 6B + AC.

If A = `[(1, 2),(2, 3)], B = [(1, 2),(3, 1)], C = [(3, 1),(1, 3)]`, find C(B − A).

 If A = `[(4, -4),(-3, 3)], B = [(2, 3),(-1, -2)], C = [(6, 5),(3, 0)]`, find AB.

If A = `[(4, -4),(-3, 3)], B = [(2, 3),(-1, -2)], C = [(6, 5),(3, 0)]`, find AC.

If A = `[(4, -4),(-3, 3)], B = [(2, 3),(-1, -2)], C = [(6, 5),(3, 0)]`, find if AB = AC.

If A = `[(4, 2),(-1, 1)]`, show that (A − 2I) (A − 3J) = 0.

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

1. A(BC)
2. (AB)C
3. If A(BC) = (AB)C

If A = `[(1, 0),(0, -1)]`, show that A³ = A.

If A = `[(3, 1),(-1, 2)]`, find A² − 5A.

If A = `[(6, 5),(7, 6)]`, show that A² − 12A + I = 0.

Answer the following question:

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

Show that `[(1, 2),(2, 1)]` is a solution of the matrix equation X² – 2X – 3I = 0, where I is the unit matrix of order 2.

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

Find x and y if `[(2x, x),(y, 3y)] [(3),(2)] = [(16),(9)]`

If A = `[(2, -3),(x, y)]` and A² = I, find x and y.

If A = `[(p, 0),(0, 2)], B = [(0, -q),(1, 0)], C = [(2, -2),(2, 2)]` and BA = C², find the values of p and q.

Find the matrix M such that `[(5, -7),(-2, 3)] M = [(-16, -6),(7, 2)]`.

If A = `[(2, 12),(0, 1)], B = [(4, x),(0, 1)]` and A² = B, find the value of x.

If A = `[(1, -1),(2, 3)], C = [(2, 3),(1, - 1)]`, find matrix B such that BА = С.

If A = `[(2, 3),(0, -1)], B = [(-8),(8)]`, find a matrix M such that 2AM = B.

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

Given `[(2, 1),(-3,4)]` . X = `[(7),(6)]`. Write:

1. the order of the matrix X.
2. the matrix X.

If `[(-1, 0),(2, 5)] x = [(-2),(9)]`, write

1. the order of matrix X
2. the matrix X

If A = `[(2, 3),(1, 2)]` find x and y so that A² – xA + yI

If A = `[(3, 2),(1, 1)]` and A² + Ax + yI = 0, find the values of x and y.

A = `[(x, 0),(1, 1)], B = [(4, 0),(y, 1)]` and C = `[(4, 0),(x, 1)]`, find the value of x and y, if AB = C.

If `[(1, 3),(2, -5)], B = [(2),(-3)]` then the order of matrix B is ______.

If 2 `[(x, 7),(9, y - 5)] + [(6, -7),(4, 5)] = [(10, 7),(22, 15)]`, then the value of x is ______.

If A = `[(6, 0),(8, -4)]` and A + 2B = 0, then matrix B is ______.

If A = `[(2, -1),(2, 0)], B = [(-3, 2),(4, 0)], C = [(1, 0),(0, 2)]` and A + D= 2B + C, then matrix D is ______.

If X + Y = `[(3, 0),(1, 5)]` and X − Y = `[(5, 0),(9, 5)]` then matrix Y is ______.

If the order of matrices A and B are 3 × 2 and 2 × 3 respectively, then the order of BA will be ______.

If A = `[(2, -3),(x, y)]` and A2 = I, then the value of y is ______.

A = `[(2, 0),(-1, 7)], I = [(1, 0),(0, 1)]` and A² = 9A + kI, then the value of k is ______.

If A= `[(-4, 2),(5, -1)], B = [(17, -1),(47, -13)]` and CA = B, then matrix C is ______.

If A = `[(1, -2),(2, 1)], B = [(3, 1),(-2, 2)]`, then (A + B) (A − B) is equal to ______.

If matrix A = `[(2, 2),(0, 2)]` and A² = `[(4, x),(0, 4)]`, then the value of x is ______.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If order of matrix A is 2 × 3 and order of matrix B is 3 × 2, then order of matrix AB will be 3 × 3.
2. If X +Y = `[(5, 2),(0, 9)]` and Y = `[(1, -2),(0, 5)]`, then x = `[(4, 4),(0, 4)]`.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. A matrix has 4 rows and 3 columns. It consists of 12 elements.
2. If `[(2x + y, x - y),(x - z, x + y + z)] = [(10, -1),(2, 8)]`, then x = 3

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If A= `[(1, 3),(2, 1),(3, -1)], B = [(2, 1),(1, 2),(1, 1)]` and A + B + C = 0, then C = `[(-3, -4),(-3, -3),(-4, 0)]`
2. If A = `[(3, 1),(-1, 2)]`, then A² = `[(9, 1),(1, 4)]`

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If A = `[(2, 0),(-1, 7)], I = [(1, 0),(0, 1)]` and A² = 9A + mI, then m = 14.
2. If A = `[(4, 0),(1, 2)], B = [(2),(3)]` and AX = B, then order of matrix X will be 2 × 2.

Let A = `[(1, 0),(2, 1)]`, B = `[(2, 3),(-1, 0)]`, Find A² + AB + B².

If A = `[(1, 2),(2, 3)] "and B" = [(2, 1),(3, 2)], "C" = [(1, 3),(3, 1)]` find the matrix C(B – A).

If A = `[(1, 2),(3, 4)] "and B" = [(2, 1),(4, 2)], "C" = [(5, 1),(7, 4)]`, compute A(B + C).

If A= `[(1, -2),(2, -1)]` and B = `[(3, 2),(-2, 1)]`, find 2B − A².

If A = `[(1, 2),(3, 4)], B = [(6, 1),(1, 1)]` and C = `[(-2, -3),(0, 1)]`, find each of the following and state if they are equal:

1. CA + В
2. A + СВ

If A = `[(-1, 3),(2, 4)], B = [(2, -3),(-4, -6)]` find the matrix AB + BA.

Evaluate:

`[(4 sin 30°, 2 cos 60°),(sin 90°, 2 cos 0°)] [(4, 5),(5, 4)]`

Given the matrices:

A = `[(2, 1),(4, 2)]`, B = `[(3, 4),(-1, -2)]` and C = `[(-3, 1),(0, -2)]`. Find: 

1. ABC
2. ACB.
   State whether ABC = ACB.

If A = `[(1, 2),(2, 1)] "and B" = [(2, 1),(1, 2)]`, find A(BA).

Given A = `[(1, 1),(8, 3)]` evaluate A² − 4A.