Chapter 10

If the matrix \(\left[\begin{array}{lll}0 & a & 5 \\ 3 & 0 & b \\ c & 2 & 0\end{array}\right]\) is skew-symmetric, then find \(a, b, c\).

If \(A\) is a skew-symmetric matrix, then find trace of \(A\).

If for a square matrix \(A=\left[a_{i j}\right] a_{i j}=i^{2}-j^{2}\), then show that \(A\) is a skew-symmetric matrix.

Let \(A\) be a square matrix, then prove that i. \(A+A^{T}\) is a symmetric matrix ii. \(A-A^{T}\) is a skew-symmetric matrix iii. \(A A^{T}\) and \(A^{T} A\) are symmetric matrices.

Prove that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skewsymmetric matrix.

If \(A\) and \(B\) are symmetric matrices, then show that \(A B\) is symmetric iff \(A B=B A\).

Show that the matrix \(B^{T} A B\) is symmetric or skew-symmetric according as \(A\) is symmetric or skewsymmetric.

Show that all natural powers of a symmetric matrix are symmetric.

Show that odd natural powers of a skew-symmetric matrix are skew-symmetric and even natural powers of a skew-symmetric matrix are symmetric.

If \(A\) and \(B\) be symmetric matrices of the same order, then show that i. \(A+B\) is a symmetric matrix ii. \(A B-B A\) is a skew-symmetric matrix iii. \(A B+B A\) is a symmetric matrix

If \(A\) and \(B\) be symmetric matrices of the same order, then show that i. \(A+B\) is a symmetric matrix ii. \(A B-B A\) is a skew-symmetric matrix iii. \(A B+B A\) is a symmetric matrix

If \(A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 2 & 3 & -1 \\ -3 & 1 & 2\end{array}\right]\), find det \(A .\) Ans. 42\(\\}\)

For what value of \(k,\left[\begin{array}{ccc}1 & 2 & -3 \\ 5 & 0 & k \\ 1 & -1 & 1\end{array}\right]\) is a singular matrix.

If \(A\) and \(B\) are two square matrices of order 3 such that \(|A|=-1,|B|=3\), then find \(|3 A B|\).

Prove that a skew-symmetric matrix of odd order must be a singular matrix.

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

Find the adjoint of \(\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3\end{array}\right]\).

Find the adjoint of the matrix \(A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 3 & 4 & 5 \\ 0 & -6 & -7\end{array}\right]\) and verify that i. \(\quad A(a d j A)=|A| I_{3}=(a d j A) A\) ii. \(\quad|a d j A|=|A|^{2}\) iii. \(a d j A^{T}=(a d j A)^{T}\) iv. \(\operatorname{adj}(\operatorname{adj} A)=|A| A\).

If \(A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 2 & 3 & -1 \\ -3 & 1 & 2\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 1 & 2 \\\ 1 & 2 & 0\end{array}\right]\), then verify that \(\operatorname{adj} A B=(\operatorname{adjB})(\) adj \(A)\).

If \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) and \(A(a d j A)=k\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]\), then find the value of \(k .\\{\) Ans. 1\(\\}\)

Find \(\left[\begin{array}{cc}1 & 3 \\ 3 & 10\end{array}\right]^{-1}\)

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

Find \(A^{-1}\), if the matrix \(A\) is given by \(A=\left[\begin{array}{cc}0.8 & 0.6 \\ -0.6 & 0.8\end{array}\right]\)

For what value of \(k\) the matrix \(A=\left[\begin{array}{cc}2 & k \\ 3 & 5\end{array}\right]\) has no inverse.

If \(A=\left[\begin{array}{cc}2 x & 0 \\ x & x\end{array}\right]\) and \(A^{-1}=\left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right]\), then find the value of \(x\).

If \(A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]\) be such that \(A^{-1}=k A\), then find the value of \(k\).

\(X\) is an unknown square matrix satisfying the equation \(\left[\begin{array}{cc}1 & 3 \\ 0 & 1\end{array}\right] X=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\). Determine the matrix \(X\).

If \(\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] A\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\), then find the matrix \(A\).

Find the inverse of the matrix \(\left.A=\mid \begin{array}{ccc}1 & 2 & -2 \\\ -1 & 3 & 0 \\ 0 & -2 & 1\end{array}\right]\). Verify that \(\left|A^{-1}\right|=\frac{1}{|A|}\).

If \(A=\left[\begin{array}{ll}2 & 5 \\ 1 & 6\end{array}\right]\), find \(A^{-1}\) and verify that \(A^{-1}=-\frac{1}{7} A+\frac{8}{7} I\)

If \(A=\left[\begin{array}{ll}2 & 0 \\ 3 & 1\end{array}\right]\) and \(B=\left[\begin{array}{ll}0 & 1 \\ 2 & 4\end{array}\right]\). Verify that \((A B)^{-1}=B^{-1} A^{-1}\).

Find the inverse of the matrix \(\left.A=\mid \begin{array}{lll}2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2\end{array}\right]\) and verify that \(A^{-1} A=I\)

Find the inverse of the matrix \(A=\left[\begin{array}{ccc}2 & 1 & 1 \\ -3 & 0 & 1 \\ -1 & 1 & 2\end{array}\right]\) and verify that \(A^{-1} A=I\).

Find the inverse of the matrix \(A=\left[\begin{array}{ccc}2 & -2 & 4 \\ 2 & 3 & 2 \\ -1 & 1 & -1\end{array}\right]\) and verify that \(A^{-1} A=I\).

If \(F(\alpha)=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]\) then show that \(F(x) F(y)=F(x+y)\). Hence prove that \([F(x)]^{-1}=F(-x)\).

If \(A=\left[\begin{array}{ccc}2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{array}\right]\), prove that \(A^{-1}=A^{2}-6 A+11 I\).

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

If \(A=\left[\begin{array}{ll}4 & 5 \\ 2 & 1\end{array}\right]\), then show that \(A-3 I=2 I+6 A^{-1}\).

Let \(A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]\), prove that \(A^{2}-4 A-5 I=O .\) Hence obtain \(A^{-1}\).

If \(\left[\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right]\left[\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right]^{-1}=\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]\), then show that \(a=\cos 2 \theta, b=\sin 2 \theta .\)

Show that the inverse of a diagonal matrix is a diagonal matrix.

Show that if \(A\) is an orthogonal matrix, then \(A^{T}\) is also orthogonal.

If \(A, B\) be orthogonal matrices, then prove that \(A B\) and \(B A\) are also orthogonal matrices.

Prove that the matrix \(A=\left|\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\\ 1 & -2 & -3\end{array}\right|\) is idempotent.

Find all idempotent diagonal matrices of order 3 .

If \(B\) is idempotent, show that \(A=I-B\) is also idempotent and that \(A B=B A=O\).

If \(A\) and \(B\) are idempotent and \(A\) and \(B\) commute, then show that \(A B\) is also idempotent.

If \(A\) and \(B\) are idempotent and \(A B=B A=O\), then show that \(A+B\) is also idempotent.

Show that \(\left[\begin{array}{rr}-6 & 5 \\ -7 & 6\end{array}\right]\) is an involutary matrix.

Show that \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]\) is an involutary matrix.