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) \(BA\)
(i) \(A + B = \begin{bmatrix} 3 & 7 \\ 1 & 7 \end{bmatrix}\)
(ii) \(A - B = \begin{bmatrix} 1 & 1 \\ 5 & -3 \end{bmatrix}\)
(iii) \(3A - C = \begin{bmatrix} 8 & 7 \\ 6 & 2 \end{bmatrix}\)
(iv) \(AB = \begin{bmatrix} -6 & 26 \\ -1 & 19 \end{bmatrix}\)
(v) \(BA = \begin{bmatrix} 11 & 10 \\ 11 & 2 \end{bmatrix}\)
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) \(\begin{bmatrix} \cos^2 x & \sin^2 x \\ \sin^2 x & \cos^2 x \end{bmatrix} + \begin{bmatrix} \sin^2 x & \cos^2 x \\ \cos^2 x & \sin^2 x \end{bmatrix}\)
(i) \(\begin{bmatrix} 2a & 2b \\ 0 & 2a \end{bmatrix}\)
(ii) \(\begin{bmatrix} (a+b)^2 & (b+c)^2 \\ (a-c)^2 & (a-b)^2 \end{bmatrix}\)
(iii) \(\begin{bmatrix} 11 & 11 & 0 \\ 16 & 5 & 21 \\ 5 & 10 & 9 \end{bmatrix}\)
(iv) \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\)
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) \(\begin{bmatrix} 3 & -1 & 3 \\ -1 & 0 & 2 \end{bmatrix}\begin{bmatrix} 2 & -3 \\ 1 & 0 \\ 3 & 1 \end{bmatrix}\)
(i) \(\begin{bmatrix} a^2+b^2 & 0 \\ 0 & a^2+b^2 \end{bmatrix}\)
(ii) \(\begin{bmatrix} 2 & 3 & 4 \\ 4 & 6 & 8 \\ 6 & 9 & 12 \end{bmatrix}\)
(iii) \(\begin{bmatrix} -3 & -4 & 1 \\ 8 & 13 & 9 \end{bmatrix}\)
(iv) \(\begin{bmatrix} 14 & 0 & 42 \\ 18 & -1 & 56 \\ 22 & -2 & 70 \end{bmatrix}\)
(v) \(\begin{bmatrix} 1 & 2 & 3 \\ 1 & 4 & 5 \\ -2 & 2 & 0 \end{bmatrix}\)
(vi) \(\begin{bmatrix} 14 & -6 \\ 4 & 5 \end{bmatrix}\)
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 \(A + (B - C) = (A + B) - C\).
\(A + B = \begin{bmatrix} 4 & 1 & -1 \\ 9 & 2 & 7 \\ 3 & -1 & 4 \end{bmatrix}\)
\(B - C = \begin{bmatrix} -1 & -2 & 0 \\ 4 & -1 & 3 \\ 1 & 2 & 0 \end{bmatrix}\)
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 \(3A - 5B\).
\(3A - 5B = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)
Simplify
\(\cos\theta\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} + \sin\theta\begin{bmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{bmatrix}\).
\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
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 \(3X + 2Y = \begin{bmatrix} 2 & -2 \\ -1 & 5 \end{bmatrix}\)
(i) \(X = \begin{bmatrix} 5 & 0 \\ 1 & 4 \end{bmatrix},\; Y = \begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix}\)
(ii) \(X = \begin{bmatrix} \frac{2}{5} & -\frac{12}{5} \\ -\frac{11}{5} & 3 \end{bmatrix},\; Y = \begin{bmatrix} \frac{2}{5} & \frac{13}{5} \\ \frac{14}{5} & -2 \end{bmatrix}\)
Find \(X\), if \(Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\) and \(2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}\).
\(X = \begin{bmatrix} -1 & -1 \\ -2 & -1 \end{bmatrix}\)
Find \(x\) and \(y\), if
\(2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}\).
\(x = 3,\; y = 3\)
Solve the equation for \(x, y, z\) and \(t\), if
\(2\begin{bmatrix} x & z \\ y & t \end{bmatrix} + 3\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3\begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}\).
\(x = 3,\; y = 6,\; z = 9,\; t = 6\)
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 \(y\).
\(x = 3,\; y = -4\)
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 \(w\).
\(x = 2,\; y = 4,\; z = 1,\; w = 3\)
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
\[ A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix} \]
\(A^2 - 5A + 6I = \begin{bmatrix} 1 & -1 & -3 \\ -1 & -1 & -10 \\ -5 & 4 & 4 \end{bmatrix}\)
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 \(A^2 = kA - 2I\).
\(k = 1\)
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) \(\text{₹}2000\)
(a) \(\text{₹}15000,\; \text{₹}15000\)
(b) \(\text{₹}5000,\; \text{₹}25000\)
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.
\(\text{₹}20160\)
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) \(k = 2,\; p = 3\)
(A)
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) \(p \times n\)
(B)