Chapter 10

Find \(x\) and \(y\) if \(\left[\begin{array}{cc}x+y & 2 \\ 1 & x-y\end{array}\right]=\left[\begin{array}{cc}3 & 2 \\ 1 & 7\end{array}\right]\)

Find \(x, y, z, w\) if \(\left[\begin{array}{cc}x-y & 2 x+z \\ 2 x-y & 3 z+w\end{array}\right]=\left[\begin{array}{cc}-1 & 5 \\ 0 & 13\end{array}\right]\)

Find \(x, y, z\) if \(\left[\begin{array}{cc}x+y & y-z \\ z-2 x & y-x\end{array}\right]=\left[\begin{array}{cc}3 & -1 \\ 1 & 1\end{array}\right]\)

Find \(a, b, c, d\) if \(\left[\begin{array}{ll}a+3 & 2 b-8 \\ c+1 & 4 d-6\end{array}\right]=\left[\begin{array}{cc}0 & -6 \\ -3 & 2 d\end{array}\right]\)

If \(A=\left[\begin{array}{ccc}2 & 3 & 1 \\ 0 & -1 & 5\end{array}\right]\) and \(B=\left[\begin{array}{ccc}1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right]\) evaluate \(3 A-4 B\).

If \(P=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 5 & 7 \\ 6 & 8 & 9\end{array}\right], Q=\left[\begin{array}{lll}2 & 0 & 3 \\ 3 & 0 & 5 \\ 5 & 7 & 0\end{array}\right]\), evaluate \(2 P-3 Q\).

If \(A=\left[\begin{array}{cc}1 & -2 \\ 3 & 0\end{array}\right], B=\left[\begin{array}{cc}-1 & 4 \\ 2 & 3\end{array}\right], C=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\), then find \(5 A-3 B+2 C .\)

If \(A=\left[\begin{array}{cc}1 & 2 \\ -1 & 0 \\ 1 & -3\end{array}\right], B=\left[\begin{array}{cc}4 & 5 \\ -1 & 0 \\ 2 & 1\end{array}\right], C=\left[\begin{array}{cc}-1 & -2 \\ -1 & 2 \\ -1 & -2\end{array}\right]\), find \(A-2 B+3 C\).

If \(\left[\begin{array}{ll}x & 0 \\ 1 & y\end{array}\right]+\left[\begin{array}{cc}-2 & 1 \\ 3 & 4\end{array}\right]=\left[\begin{array}{ll}3 & 5 \\ 6 & 3\end{array}\right]-\left[\begin{array}{ll}2 & 4 \\ 2 & 1\end{array}\right]\), then find \(x, y .\)

Given \(A=\left[\begin{array}{ccc}1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1\end{array}\right]\) and \(B=\left[\begin{array}{ccc}3 & -1 & 2 \\ 4 & 2 & 5 \\\ 2 & 0 & 3\end{array}\right]\). Find the matrix \(C\) such that \(A+2 C=B\).

Find \(A\) and \(B\) if \(A+B=\left[\begin{array}{cc}1 & -2 \\ 3 & 4\end{array}\right]\) and \(A-B=\left[\begin{array}{cc}3 & 2 \\ -1 & 0\end{array}\right]\).

If \(A+B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\) and \(A-2 B=\left[\begin{array}{cc}-1 & 1 \\ 0 & -1\end{array}\right]\), then find \(A\).

Find \(A\) and \(B\) if \(2 A-B=\left[\begin{array}{ccc}3 & -3 & 0 \\ 3 & 3 & 2\end{array}\right]\) and \(2 B+A=\left[\begin{array}{ccc}4 & 1 & 5 \\ -1 & 4 & -4\end{array}\right]\).

If \(I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], J=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\) and \(B=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]\), then show that \(B=I \cos \theta+J \sin \theta\).

What is the order of \(\left[\begin{array}{lll}x & y & z\end{array}\right]\left[\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]\)

Given \(A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 1 & 2 & -1\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 1 & 2 \\ 2 & 0 & 1 \\ 1 & 2 & 0\end{array}\right]\), find \(A B\) and \(B A\).

Given \(A=\left[\begin{array}{ll}1 & 2\end{array}\right]\) and \(B=\left[\begin{array}{l}3 \\ 1\end{array}\right]\), find \(A B\) and \(B A\).

If \(A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right], B=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\), then find \(A B\) and \(B A\) and show that \(A B \neq B A\).

If \(A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 2 & 3 & -1 \\ -3 & 1 & 2\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 1 & 2 \\\ 1 & 2 & 0\end{array}\right]\), obtain the product \(A B\) and \(B A\) and show that \(A B \neq B A\).

If \(A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\), then find \(A^{2}\) and \(A^{3}\).

If \(A_{1 \times 2}=\left[\begin{array}{ll}a & b\end{array}\right], B_{1 \times 2}=\left[\begin{array}{ll}-b & -a\end{array}\right]\) and \(C_{2 \times 1}=\left[\begin{array}{c}a \\ -a\end{array}\right]\), then show that \(A C=B C\).

Given \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]\) and \(B=\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\), find \(A B\).

Given \(A=\left[\begin{array}{ll}1 & 0\end{array}\right]\) and \(B=\left[\begin{array}{l}0 \\ 1\end{array}\right]\), find \(A B\) and \(B A\).

If \(A=\left[\begin{array}{cc}2 & -1 \\ 0 & 1\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & 0 \\ -1 & -1\end{array}\right]\), show that \((A+B)^{2}=A^{2}+A B+B A+B^{2} \neq A^{2}+2 A B+B^{2}\)

If \(A=\left[\begin{array}{cc}-1 & 2 \\ 2 & 3\end{array}\right], B=\left[\begin{array}{ll}3 & 0 \\ 1 & 1\end{array}\right]\). Verify that \((A+B)^{2}=A^{2}+A B+B A+B^{2} \neq A^{2}+2 A B+B^{2}\).

If \(A=\left[\begin{array}{cc}0 & 1 \\ 1 & 1\end{array}\right], B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]\), show that \((A+B)(A-B) \neq A^{2}-B^{2} .\)

If \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right], C=\left[\begin{array}{ll}5 & 1 \\ 7 & 4\end{array}\right]\). Show that \(A(B+C)=A B+A C\).

The matrix \(\boldsymbol{R}(t)\) is defined by \(R(t)=\left[\begin{array}{cc}\cos t & \sin t \\ -\sin t & \cos t\end{array}\right]\). Show that \(\boldsymbol{R}(\mathbf{s}) \boldsymbol{R}(\boldsymbol{t})=\boldsymbol{R}(\mathbf{s}+\boldsymbol{t})\).

If \(A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right], B=\left[\begin{array}{cc}\cos \beta & -\sin \beta \\\ \sin \beta & \cos \beta\end{array}\right]\). Show that \(A B=B A\).

If \(a, b, c, d\) are real numbers and \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\), prove that \(A^{2}-(a+d) A+(a d-b c) I=O\).

Show that \(E^{2} F+F^{2} E=E\), where \(E=\left[\begin{array}{lll}0 & 0 & 1 \\\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right], F=\left[\begin{array}{lll}1 & 0 & 0 \\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\).

Find \(x\) so that \(\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{lll}1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 3 & 2\end{array}\right]\left[\begin{array}{l}1 \\ 1 \\ x\end{array}\right]=O\).

If \(A=\left[\begin{array}{cc}0.8 & 0.6 \\ -0.6 & 0.8\end{array}\right]\), find \(A^{3} .\)

If \(X=\left[\begin{array}{cc}3 & -4 \\ 1 & -1\end{array}\right]\), then find \(X^{3}\).

If \(A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\), then find \(A^{11}\).

If \(J_{1}=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\) and \(J_{2}=\left[\begin{array}{cc}1 & 0 \\ -1 & 0\end{array}\right]\), then find \(J_{1}^{2}+J_{2}^{2}\).

If \(A=\left[\begin{array}{ccc}1 & -3 & 2 \\ 2 & 1 & -3 \\ 4 & -3 & -1\end{array}\right], B=\left[\begin{array}{cc}1 & 4 \\ 2 & 1 \\ 1 & -2\end{array}\right]\) and \(C=\left[\begin{array}{ccc}2 & 1 & -1 \\ 3 & -2 & -1\end{array}\right]\). Show that \((A B) C=A(B C)\).

If \(f(x)=x^{2}-5 x+6 I\), find \(f(A)\) if \(A=\left[\begin{array}{ccc}2 & 0 & 1 \\\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right]\).

If \(A=\left[\begin{array}{ll}2 & 4 \\ 4 & 3\end{array}\right], X=\left[\begin{array}{l}n \\ 1\end{array}\right], B=\left[\begin{array}{c}8 \\\ 11\end{array}\right]\) and \(A X=B\), then find \(n\).

If \(A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\) and \(A^{2}=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]\), then show that \(\alpha=a^{2}+b^{2}, \beta=2 a b\).

If \(A=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]\) and \(A^{2}-4 A-n I_{2}=O\), then find the value of \(n\).

If \(A=\left[\begin{array}{ll}1 & 3 \\ 3 & 4\end{array}\right]\) and \(A^{2}-k A-5 I_{2}=O\), then find the value of \(k\).

If \(A=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right], I\) is the unit matrix of order 2 and \(a, b\) are arbitrary constants, then show that \((a I+b A)^{2}=a^{2} I+2 a b A\)

If \(A=\left[\begin{array}{cc}1 & -2 \\ 5 & 3\end{array}\right]\), then find \(A+A^{T}\).

Verify that \((A B)^{T}=B^{T} A^{T} \neq A^{T} B^{T}\), where \(A=\left[\begin{array}{lll}1 & 1 & 1 \\ 2 & 2 & 3 \\ 2 & 4 & 9\end{array}\right], B=\left[\begin{array}{ccc}0 & 1 & -1 \\ -3 & 2 & 4 \\ 1 & 1 & 0\end{array}\right]\).

If \(A=\left[\begin{array}{ccc}4 & -2 & 3 \\ -4 & 2 & 5\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & 3 \\ -1 & 0 \\ 2 & 4\end{array}\right]\), show that \((A B)^{T}=B^{T} A^{T} .\)

If \(A=\left[\begin{array}{ccc}1 & -1 & 0 \\ 2 & 1 & 3 \\ 4 & 1 & 8\end{array}\right]\) and \(B=\left[\begin{array}{ccc}4 & 1 & 0 \\ 2 & -3 & 1 \\\ 1 & 1 & -1\end{array}\right]\), then verify that \((A B)^{T}=B^{T} A^{T}\).

If matrix \(A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]\), where \(a, b, c\) are real positive numbers, \(a b c=1\) and \(A^{T} A=I\), then find the value of \(a^{3}+b^{3}+c^{3}\).

If \(A=\left[\begin{array}{ll}5 & x \\ y & 0\end{array}\right]\) is a symmetric matrix, then show that \(x=y\).

If \(A=\left[\begin{array}{cc}4 & x+2 \\ 2 x-3 & x+1\end{array}\right]\) is symmetric, then find \(x\).