NCERT Solutions
Class 12 - Mathematics Part-1

Chapter 3: MATRICES

In the matrix

\[ A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix} \]

write:

(i) The order of the matrix

(ii) The number of elements

(iii) Write the elements \(a_{13}, a_{21}, a_{33}, a_{24}, a_{23}\).

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 \(2 \times 2\) matrix \(A = [a_{ij}]\), whose elements are given by:

(i) \(a_{ij} = \frac{(i + j)^2}{2}\)

(ii) \(a_{ij} = \frac{i}{j}\)

(iii) \(a_{ij} = \frac{(i + 2j)^2}{2}\)

Construct a \(3 \times 4\) matrix, whose elements are given by:

(i) \(a_{ij} = \frac{1}{2} | -3i + j |\)

(ii) \(a_{ij} = 2i - j\)

Find the values of \(x, y\) and \(z\) from the following equations:

(i) \(\begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}\)

(ii) \(\begin{bmatrix} x + y & 2 \\ 5 + z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}\)

(iii) \(\begin{bmatrix} x + y + z \\ x + z \\ y + z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}\)

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

\[ \begin{bmatrix} a - b & 2a + c \\ 2a - b & 3c + d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix} \]

Let

\[ A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix},\; B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix},\; C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix} \]

Find each of the following:

(i) \(A + B\)

(ii) \(A - B\)

(iii) \(3A - C\)

(iv) \(AB\)

(v) \(BA\)

Compute the following:

(i) \(\begin{bmatrix} a & b \\ -b & a \end{bmatrix} + \begin{bmatrix} a & b \\ b & a \end{bmatrix}\)

(ii) \(\begin{bmatrix} a^2+b^2 & b^2+c^2 \\ a^2+c^2 & a^2+b^2 \end{bmatrix} + \begin{bmatrix} 2ab & 2bc \\ -2ac & -2ab \end{bmatrix}\)

(iii) \(\begin{bmatrix} -1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5 \end{bmatrix} + \begin{bmatrix} 12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4 \end{bmatrix}\)

(iv) \(\begin{bmatrix} \cos^2 x & \sin^2 x \\ \sin^2 x & \cos^2 x \end{bmatrix} + \begin{bmatrix} \sin^2 x & \cos^2 x \\ \cos^2 x & \sin^2 x \end{bmatrix}\)

Compute the indicated products.

(i) \(\begin{bmatrix} a & b \\ -b & a \end{bmatrix}\begin{bmatrix} a & b \\ b & a \end{bmatrix}\)

(ii) \(\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\begin{bmatrix} 2 & 3 & 4 \end{bmatrix}\)

(iii) \(\begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix}\)

(iv) \(\begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix}\begin{bmatrix} 1 & -3 & 5 \\ 0 & 2 & 4 \\ 3 & 0 & 5 \end{bmatrix}\)

(v) \(\begin{bmatrix} 2 & 1 \\ 3 & 2 \\ -1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 1 \\ -1 & 2 & 1 \end{bmatrix}\)

(vi) \(\begin{bmatrix} 3 & -1 & 3 \\ -1 & 0 & 2 \end{bmatrix}\begin{bmatrix} 2 & -3 \\ 1 & 0 \\ 3 & 1 \end{bmatrix}\)

If

\[ A = \begin{bmatrix} 1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1 \end{bmatrix},\; B = \begin{bmatrix} 3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3 \end{bmatrix},\; C = \begin{bmatrix} 4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3 \end{bmatrix} \]

then compute \((A + B)\) and \((B - C)\). Also, verify that \(A + (B - C) = (A + B) - C\).

If

\[ A = \begin{bmatrix} \frac{2}{3} & 1 & \frac{5}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ \frac{7}{3} & 2 & \frac{2}{3} \end{bmatrix},\; B = \begin{bmatrix} \frac{2}{5} & \frac{3}{5} & 1 \\ \frac{1}{5} & \frac{2}{5} & \frac{4}{5} \\ \frac{7}{5} & \frac{6}{5} & \frac{2}{5} \end{bmatrix} \]

then compute \(3A - 5B\).

Simplify

\(\cos\theta\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} + \sin\theta\begin{bmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{bmatrix}\).

Find \(X\) and \(Y\), if

(i) \(X + Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}\) and \(X - Y = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\)

(ii) \(2X + 3Y = \begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix}\) and \(3X + 2Y = \begin{bmatrix} 2 & -2 \\ -1 & 5 \end{bmatrix}\)

Find \(X\), if \(Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\) and \(2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}\).

Find \(x\) and \(y\), if

\(2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}\).

Solve the equation for \(x, y, z\) and \(t\), if

\(2\begin{bmatrix} x & z \\ y & t \end{bmatrix} + 3\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3\begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}\).

If \(x\begin{bmatrix} 2 \\ 3 \end{bmatrix} + y\begin{bmatrix} -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 5 \end{bmatrix}\), find the values of \(x\) and \(y\).

Given

\(3\begin{bmatrix} x & y \\ z & w \end{bmatrix} = \begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix} + \begin{bmatrix} 4 & x + y \\ z + w & 3 \end{bmatrix}\), find the values of \(x, y, z\) and \(w\).

If

\[ F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

show that \(F(x)\,F(y) = F(x + y)\).

Show that

(i) \(\begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix}\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} \ne \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix}\)

(ii) \(\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix}\begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{bmatrix} \ne \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix}\)

Find \(A^2 - 5A + 6I\), if

\[ A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix} \]

If

\[ A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \]

prove that \(A^3 - 6A^2 + 7A + 2I = 0\).

If

\[ A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix},\; I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

find \(k\) so that \(A^2 = kA - 2I\).

If

\[ A = \begin{bmatrix} 0 & -\tan\left(\frac{\alpha}{2}\right) \\ \tan\left(\frac{\alpha}{2}\right) & 0 \end{bmatrix} \]

and \(I\) is the identity matrix of order 2, show that

\[ I + A = (I - A)\begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}. \]

A trust fund has \(\text{₹}30000\) 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 \(\text{₹}30000\) among the two types of bonds. If the trust fund must obtain an annual total interest of:

(a) \(\text{₹}1800\)

(b) \(\text{₹}2000\)

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are \(\text{₹}80\), \(\text{₹}60\) and \(\text{₹}40\) 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 \(2 \times n,\; 3 \times k,\; 2 \times p,\; n \times 3\) and \(p \times k\), respectively. Choose the correct answer.

The restriction on \(n, k\) and \(p\) so that \(PY + WY\) will be defined are:

(A) \(k = 3,\; p = n\)

(B) \(k\) is arbitrary, \(p = 2\)

(C) \(p\) is arbitrary, \(k = 3\)

(D) \(k = 2,\; p = 3\)

Assume \(X, Y, Z, W\) and \(P\) are matrices of order \(2 \times n,\; 3 \times k,\; 2 \times p,\; n \times 3\) and \(p \times k\), respectively. Choose the correct answer.

If \(n = p\), then the order of the matrix \(7X - 5Z\) is:

(A) \(p \times 2\)

(B) \(2 \times n\)

(C) \(n \times 3\)

(D) \(p \times n\)

Find the transpose of each of the following matrices:

(i) \(\begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}\)

(ii) \(\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}\)

(iii) \(\begin{bmatrix} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{bmatrix}\)

If

\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix} \]

then verify that

(i) \((A + B)' = A' + B'\)

(ii) \((A - B)' = A' - B'\)

If

\[ A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix} \]

then verify that

(i) \((A + B)' = A' + B'\)

(ii) \((A - B)' = A' - B'\)

If

\[ A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix} \]

then find \((A + 2B)'\).

For the matrices \(A\) and \(B\), verify that \((AB)' = B'A'\), where

(i) \(A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix},\; B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}\)

(ii) \(A = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix},\; B = \begin{bmatrix} 1 & 5 & 7 \end{bmatrix}\)

If

(i) \(A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}\), then verify that \(A'A = I\).

(ii) If \(A = \begin{bmatrix} \sin\alpha & \cos\alpha \\ -\cos\alpha & \sin\alpha \end{bmatrix}\), then verify that \(A'A = I\).

(i) Show that the matrix

\[ A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix} \]

is a symmetric matrix.

(ii) Show that the matrix

\[ A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix} \]

is a skew symmetric matrix.

For the matrix \(A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}\), verify that

(i) \((A + A')\) is a symmetric matrix

(ii) \((A - A')\) is a skew symmetric matrix

Find \(\frac{1}{2}(A + A')\) and \(\frac{1}{2}(A - A')\), when

\[ A = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix} \]

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

(i) \(\begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}\)

(ii) \(\begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}\)

(iii) \(\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\)

(iv) \(\begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}\)

Matrices \(A\) and \(B\) will be inverse of each other only if

(A) \(AB = BA\)

(B) \(AB = BA = 0\)

(C) \(AB = 0,\; BA = I\)

(D) \(AB = BA = I\)

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

\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]

satisfy the equation \(A'A = I\).

For what values of \(x\) :

\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix}\begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = 0\)?

If \(A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\), show that \(A^2 - 5A + 7I = 0\).

Find \(x\), if

\(\begin{bmatrix} x & -5 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}\begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = 0\).

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

Market I: \(x = 10000,\; y = 2000,\; z = 18000\)

Market II: \(x = 6000,\; y = 20000,\; z = 8000\)

(a) If unit sale prices of \(x, y\) and \(z\) are \(\text{₹}2.50,\; \text{₹}1.50\) and \(\text{₹}1.00\), respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are \(\text{₹}2.00,\; \text{₹}1.00\) and 50 paise respectively. Find the gross profit.

Find the matrix \(X\) so that

\(X\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}\).

Choose the correct answer.

If \(A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}\) is such that \(A^2 = I\), then

(A) \(1 + \alpha^2 + \beta\gamma = 0\)

(B) \(1 - \alpha^2 + \beta\gamma = 0\)

(C) \(1 - \alpha^2 - \beta\gamma = 0\)

(D) \(1 + \alpha^2 - \beta\gamma = 0\)

Choose the correct answer.

If the matrix \(A\) is both symmetric and skew symmetric, then

(A) \(A\) is a diagonal matrix

(B) \(A\) is a zero matrix

(C) \(A\) is a square matrix

(D) None of these

Choose the correct answer.

If \(A\) is a square matrix such that \(A^2 = A\), then \((I + A)^3 - 7A\) is equal to

(A) \(A\)

(B) \(I - A\)

(C) \(I\)

(D) \(3A\)