Chapter 10, Problem 9T

(a)

To determine

The value of $A+B$.

Answer to Problem 9T

The matrix addition is not possible.

Explanation of Solution

The given matrices are,

$A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$

To add the matrices the number of rows and column of two matrices must be equal.

The number of rows and columns of matrix $A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$ is 2.

The number of rows and columns of matrix $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ are 3 and 2 respectively.

Thus, the addition of given matrix is not possible.

(b)

To determine

The value of $AB$.

Answer to Problem 9T

The matrix multiplication is not possible.

Explanation of Solution

The given matrices are,

$A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$

To perform multiplication operation between the matrices the significant condition is that the number of columns of first matrix must be equal to the number of rows of second matrix.

The number of columns of matrix $A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$ is 2.

The number of rows of matrix $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ is 3.

The number of columns of first matrix is not equal to number of rows of second matrix.

Thus, Multiplication of given matrix is not possible.

(c)

To determine

The value of $BA-3B$.

Answer to Problem 9T

The value of $BA-3B$ is $\left[\begin{array}{cc}6& 10\\ 3& -2\\ -3& 9\end{array}\right]$.

Explanation of Solution

The given matrices are,

$A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$

Apply matrix multiplication operation to find the value of $BA-3B$.

$\begin{array}{c}\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]\times \left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]-3\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]=\left[\begin{array}{cc}12& 22\\ 0& 1\\ 6& 9\end{array}\right]-\left[\begin{array}{cc}6& 12\\ -3& 3\\ 9& 0\end{array}\right]\\ =\left[\begin{array}{cc}6& 10\\ 3& -2\\ -3& 9\end{array}\right]\end{array}$

Thus, the value of $BA-3B$ is $\left[\begin{array}{cc}6& 10\\ 3& -2\\ -3& 9\end{array}\right]$.

(d)

To determine

The value of $CBA$.

Answer to Problem 9T

The value of $CBA$ is $\left[\begin{array}{cc}36& 58\\ 0& -3\\ 18& 28\end{array}\right]$.

Explanation of Solution

The given matrices are,

$A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$

Apply matrix multiplication operation to find the value of $CBA$.

$\begin{array}{c}\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]\times \left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]\times \left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]\times \left[\begin{array}{cc}12& 22\\ 0& 1\\ 6& 9\end{array}\right]\\ =\left[\begin{array}{cc}12+24& 22+36\\ 0& -22+1+18\\ 18& 28\end{array}\right]\\ =\left[\begin{array}{cc}36& 58\\ 0& -3\\ 18& 28\end{array}\right]\end{array}$

Thus, the value of $CBA$ is $\left[\begin{array}{cc}36& 58\\ 0& -3\\ 18& 28\end{array}\right]$.

(e)

To determine

The value of $A^{-1}$.

Answer to Problem 9T

The inverse of matrix $A$ is $\left[\begin{array}{cc}2& -\frac{3}{2}\\ -1& 1\end{array}\right]$.

Explanation of Solution

The given matrices are,

$A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$

The matrix $2\times 2$ is $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Compare the matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ with $A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, the value of a is 2, b is 3, c is 2 and d is 4.

Formula to calculate the inverse of $2\times 2$ matrix is,

$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Substitute $2$ for $a$, $3$ for $b$, $2$ for $c$ and $4$ for $d$ in above expression to find $A^{-1}$.

$\begin{array}{c}{A}^{-1}=\frac{1}{2\times 4-3\times 2}\left[\begin{array}{cc}4& -3\\ -2& 2\end{array}\right]\\ =\left[\begin{array}{cc}2& -\frac{3}{2}\\ -1& 1\end{array}\right]\end{array}$

Thus, the inverse of matrix $A$ is $\left[\begin{array}{cc}2& -\frac{3}{2}\\ -1& 1\end{array}\right]$.

(f)

To determine

The value of $B^{-1}$.

Answer to Problem 9T

The inverse of matrix $B$ is not possible.

Explanation of Solution

Given:

The given matrices are,

$A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$

The essential condition to find the inverse of the matrix is that the given matrix must be a square matrix.

The matrix $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ is not a square matrix.

Thus, the inverse of matrix is not possible.

(g)

To determine

The determinant of matrix $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$.

Answer to Problem 9T

The determinant of matrix $B$ is not possible.

Explanation of Solution

Given:

The given matrices are,

$A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$

The essential condition to find the determinant of the matrix is that the given matrix must be a square matrix.

The matrix $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ is not a square matrix.

Thus, the determinant of matrix is not possible.

(h)

To determine

The determinant of matrix $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$.

Answer to Problem 9T

The determinant of the matrix $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$ is $-3$.

Explanation of Solution

Given:

The given matrices are,

$A=\left[\begin{array}{cc}2& 3\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}2& 4\\ -1& 1\\ 3& 0\end{array}\right]$ and $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$

The determinant of the matrix $C$ is,

$\begin{array}{c}\det \left(C\right)=\left|\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right|\\ =\left|\begin{array}{cc}1& 2\\ 1& 3\end{array}\right|-0\left|\begin{array}{cc}-1& 2\\ 0& 3\end{array}\right|+4\left|\begin{array}{cc}-1& 1\\ 0& 1\end{array}\right|\\ =\left(3-2\right)+4\left(-1\right)\\ =-3\end{array}$

The determinant of the matrix $C=\left[\begin{array}{ccc}1& 0& 4\\ -1& 1& 2\\ 0& 1& 3\end{array}\right]$ is $-3$.