Chapter 11, Problem 8CT

To determine

The value of the expression is $A-3C$ where $A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right],B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right],C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right]$.

Answer to Problem 8CT

Solution:

The value of the expression $A-3C$ is $\left[\begin{array}{cc}-11& -19\\ -3& 5\\ 6& -22\end{array}\right]$.

Explanation of Solution

Given information:

The matrices, $A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right],B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right],C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right]$.

Explanation:

Consider an expression $A-3C$,

Toperform matrix operation using following rules, If ${\left[A\right]}_{n\times n}=a_{ij}$ and ${\left[B\right]}_{n\times n}=b_{ij}$ are two matrices then $\left[A-B\right]=a_{ij}-b_{ij}$

$\alpha \left[A\right]=\alpha a_{ij}$ where $\alpha$ is any scalar.

$\Rightarrow A-3C=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right]-3\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right]$

$\Rightarrow A-3C=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right]-\left[\begin{array}{cc}12& 18\\ 3& -9\\ -3& 24\end{array}\right]$

$\Rightarrow A-3C=\left[\begin{array}{cc}1-12& -1-18\\ 0-3& -4-(-9)\\ 3-(-3)& 2-24\end{array}\right]$

$\Rightarrow A-3C=\left[\begin{array}{cc}-11& -19\\ -3& 5\\ 6& -22\end{array}\right]$

Therefore, the value of the expression $A-3C$ is $\left[\begin{array}{cc}-11& -19\\ -3& 5\\ 6& -22\end{array}\right]$.