NCERT solutions for Mathematics Part 1 and 2 [English] Class 12 chapter 3 - Matrices [Latest edition]

In the matrix A = `[(2,5,19,-7),(35,-2, 5/2 ,12),(sqrt3, 1, -5 , 17)]`, write the order of the matrix.

In the matrix A = `[(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]`, write the number of elements.

In the matrix A = `[(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]`, write the elements a₁₃, a₂₁, a₃₃, a₂₄, a₂₃.

If a matrix has 24 elements, what are the possible orders it can have? What if it has 13 elements?

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Construct a 2 × 2 matrix, A = [a_ij], whose elements are given by:

`a_(ij) = (i+j)^2/2`

Construct a 2 × 2 matrix, A = [a_ij], whose elements are given by:

`a_(ij) = i/j`

Construct a 2 × 2 matrix, A = [a_ij], whose elements are given by:

`a_(ij) = (1 + 2j)^2/2`

Construct a 3 × 4 matrix, whose elements are given by:

`a_(ij) = 1/2 |-3i + j|`

Construct a 3 × 4 matrix, whose elements are given by:

a_ij = 2i − j

Find the value of x, y, and z from the following equation:

`[(4,3),(x,5)] = [(y,z),(1,5)]`

Find the value of x, y, and z from the following equation:

`[(x+y, 2),(5+z, xy)] = [(6,2), (5,8)]`

Find the value of x, y, and z from the following equation:

`[(x+y+z), (x+z), (y+z)] = [(9),(5),(7)]`

Find the value of a, b, c, and d from the equation:

`[(a-b, 2a+c),(2a-b, 3x+d)] = [(-1,5),(0,13)]`

`A = [a_(ij)]_(mxxn)` is a square matrix, if ______.

Which of the given values of x and y make the following pair of matrices equal?

`[(3x+7, 5),(y+1, 2-3x)] = [(0,y-2),(8,4)]`

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is ______.

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

Find  A + B

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

Find A - B

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

Find  3A - C

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

Find AB

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

Find BA

Compute the following:

`[(a,b),(-b, a)] + [(a,b),(b,a)]`

Compute the following:

`[(a^2+b^2, b^2+c^2),(a^2+c^2, a^2+b^2)] + [(2ab , 2bc),(-2ac, -2ab)]`

Compute the following: 

`[(-1,4, -6),(8,5,16),(2,8,5)] + [(12,7,6),(8,0,5),(3,2,4)]`

Compute the following:

`[(cos^2x, sin^2 x),(sin^2 x ,cos^2 x)]+[(sin^2 x, cos^2 x), (cos^2 x, sin^2 x)]`

Compute the indicated product:

`[(a,b),(-b,a)][(a,-b),(b,a)]`

Compute the indicated product.

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

Compute the indicated product.

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

Compute the indicated products

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

Compute the indicated product.

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

Compute the indicated product.

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

if `A = [(1,2,-3),(5,0,2),(1,-1,1)], B = [(3,-1,2),(4,2,5),(2,0,3)] and C = [(4,1,2),(0,3,2),(1,-2,3)]` then compute (A + B) and (B - C). Also verify that A + (B -C) = (A + B) - C.

If ` A = [(2/3, 1, 5/3), (1/3, 2/3, 4/3),(7/3, 2, 2/3)]` and `B = [(2/5, 3/5,1),(1/5, 2/5, 4/5), (7/5,6/5, 2/5)]` then compute 3A - 5B.

Simplify, `cos theta[(cos theta, sintheta),(-sin theta, cos theta)] + sin theta [(sin theta, -cos theta), (cos theta, sin theta)]`

Find X and Y, if `X + Y = [(7,0),(2,5)] and X - Y = [(3,0),(0,3)]`

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

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

Find x and y, if  `2[(1,3),(0, x)]+[(y,0),(1,2)] = [(5,6),(1,8)]`

Solve the equation for x, y, z and t if `2[(x,z),(y, t)] + 3[(1,-1),(0,2)] = 3[(3,5),(4,6)]`

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

Given `3[(x,y),(z,w)] = [(x,6),(-1,2W)] + [(4,x+y),(Z+W,3)]` find the values of x, y, z and w.

If F(x) = `[(cosx, -sinx,0), (sinx, cosx, 0),(0,0,1)]`  show that F(x)F(y) = F(x + y)

Show that `[(5, -1),(6,7)][(2,1),(3,4)] != [(2,1),(3,4)][(5,-1),(6,7)]`

Show that `[(1,2,3),(0,1,0),(1,1,0)][(-1,1,0),(0,-1,1),(2,3,4)]!=[(-1,1,0),(0,-1,1),(2,3,4)][(1,2,3),(0,1,0),(1,1,0)]`

Find `A^2 - 5A + 6I if A = [(2,0,1),(2,1,3),(1,-1,0)]`

if `A = [(1,0,2),(0,2,1),(2,0,3)]` , prove that `A^2 - 6A^2 + 7A + 2I = 0`

if A = `[(3, -2),(4,-2)] and l = Matric [(1,0),(0,1)]`  find k so that `A^2 = kA - 2I`

if `A = [(0, -tan  alpha/2), (tan  alpha/2, 0)]` and I is the identity matrix of order 2, show that I + A = `(I -A)[(cos alpha, -sin alpha),(sin alpha, cos alpha)]`

A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of Rs. 1,800.

A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of Rs 2,000.

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively.

The restrictions on n, k and p so that PY + WY will be defined are ______.

Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k respectively.

If n = p, then the order of the matrix is 7X - 5Z is ______.

Find the transpose of the following matrices:

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

Find the transpose of the following matrices:

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

Find the transpose of the following matrices:

`[(-1,5,6),(sqrt3, 5, 6),(2,3,-1)]`

If `A = [(-1,2,3),(5,7,9),(-2,1,1)]  "and"  B = [(-4,1,-5),(1,2,0),(1,3,1)]` then verify that (A+ B)' = A' + B'

if `A = [(-1,2,3),(5,7,9),(-2,1,1)] and B = [(-4,1,-5),(1,2,0),(1,3,1)]` then verify that (A- B)' = A' - B'

if `A' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))]` then verify that (A + B)' = A' + B'

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

if A' = `[(-2,3),(1,2)] and B = [(-1,0),(1,2)]`  then find (A + 2B)'

For the matrices A and B, verify that (AB)′ = B'A' where `A =[(1),(-4), (3)], B = [-1, 2  1]`

For the matrices A and B, verify that (AB)′ = B'A'  where `A =[(0), (1),(2)] , B =[1 , 5, 7]`

If A = `[(cos alpha, sin alpha), (-sin alpha, cos alpha)]` then verify that  A' A = I

If A = `[(sin alpha, cos alpha), (-cos alpha, sin alpha)]` then verify that  A'A = I

Show that the matrix  A = `[(1,-1,5),(-1,2,1),(5,1,3)]` is a symmetric matrix.

Show that the matrix  A = `[(0,1,-1),(-1,0,1),(1,-1,0)]` is a skew symmetric matrix.

For the matrix A = `[(1,5),(6,7)]` verify that (A + A') is a symmetric matrix.

For the matrix A = `[(1,5),(6,7)]` verify that (A - A') is a skew symmetric matrix.

Find `1/2` (A + A')  and  `1/2` (A -A') When `A = [(0, a, b),(-a,0,c),(-b,-c,0)]`

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

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

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

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

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

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

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

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

If A, B are symmetric matrices of same order, then AB − BA is a ______.

If A= `[(cos alpha, -sin alpha), (sin alpha, cos alpha)]` then A + A' = I then the value of α is  ______.

Find the inverse of each of the matrices, if it exists. [`(1, -1),(2,3)`]

if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = `[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))]` `n in N`

Find the inverse of each of the matrices, if it exists.` [(2,1),(1,1)]`

Find the inverse of each of the matrices, if it exists.

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

if `A = [(3,-4),(1,-1)]` then prove A"=` [(1+2n, -4n),(n, 1-2n)]` where n is any positive integer

Find the inverse of each of the matrices, if it exists. 

`[(2,3),(5,7)]`

Find the inverse of each of the matrices, if it exists.

`[(2,7),(1,4)]`

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

[(3,10),(2,7)]

`Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Find the inverse of each of the matrices, if it exists.

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

Matrices A and B will be inverse of each other only if ______.

Let A = `[(0,1),(0,0)]`show that (aI+bA)ⁿ  = aⁿI + na^n-1 bA , where I is the identity matrix of order 2 and n ∈ N

If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.

Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Find the values of x, y, z if the matrix `A = [(0,2y,z),(x,y,-z),(x , -y,z)]` satisfy the equation A'A = I.

For what values of x, `[1,2,1] [(1,2,0),(2,0,1),(1,0,2)][(0),(2),(x)]` = O?

If A = `[(3,1),(-1,2)]`  show that  `A^2 - 5A + 7I = 0`.

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

A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below:

1. If the unit sale prices of x, y and z are Rs 2.50, Rs 1.50, and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
2. If the unit costs of the above three commodities are Rs 2.00, Rs 1.00, and 50 paise, respectively,. Find the gross profit.

Find the matrix X so that  X`[(1,2,3),(4,5,6)]= [(-7,-8,-9),(2,4,6)]`

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N

If A = `[(alpha, beta),(gamma, -alpha)]` is such that A2 = I then ______.

If the matrix A is both symmetric and skew symmetric, then ______.

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