Chapter 16.CR, Problem 1CR

To determine

To perform:

To perform the matrix operations for $\left[C\right]\left[B\right]-3\left[D\right]$ where

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

Answer to Problem 1CR

Solution:

$\left[\begin{array}{ccc}4& -5& 0\\ 7& -1& 2\\ -1& -2& 4\end{array}\right]$

Explanation of Solution

Approach:

Fundamental Matrix operations are equality, addition, subtraction, multiplication by a scalar and multiplication.

Equality: Two matrices $\left[A\right]$ and $\left[B\right]$ are said to be equal if and only if matrices are of same order and the corresponding entries are the same.

Addition: When two matrices $\left[A\right]$ and $\left[B\right]$ are added, the resulting matrix $\left[C\right]$ is obtained if and only if the both matrices are of same order and the entries are found by adding the corresponding entries.

Subtraction: When matrix $\left[B\right]$ is subtracted from $\left[A\right]$, the resulting matrix $\left[C\right]$ is obtained if and only if the both matrices are of same order and the entries are found by subtracting the entries of $\left[B\right]$ from the corresponding entries of $\left[A\right]$.

Multiplication by a scalar: If a matrix $\left[A\right]$ is multiplied by the scalar (real number) $x,$ the resulting matrix is the matrix in which each entry is multiplied by $x.$

Multiplication: Let $\left[A\right]$ be an $m\times r$ matrix and $\left[B\right]$ be an $r\times n$ matrix. The product of $\left[A\right]\left[B\right]=\left[C\right]$ is an $m\times n$ matrix. The entry in the $i$ th row and $j$ th column of $\left[A\right]\left[B\right]$ is the sum of the products formed by multiplying each entry of the $i$ th row of $\left[A\right]$ by the corresponding entry in the $j$ th column of $\left[B\right].$

Check whether the number of columns of $\left[A\right]$ is equal to the number of rows of $\left[B\right]$

Calculation:

Given matrix operation is $\left[C\right]\left[B\right]-3\left[D\right]$

$\left[B\right]=\left[\begin{array}{ccc}2& -1& 0\\ 1& 0& 1\end{array}\right] \left[C\right]=\left[ \begin{array}{cc}2& 0\\ 1& 2\\ -1& 1\end{array}\right] \left[D\right]=\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 1& -1\end{array}\right]$

In order to multiply two matrices, columns of first matrix should be equal to rows of the second matrix.

Check whether the number of columns of $\left[C\right]$ is equal to the number of rows of $\left[B\right]$.

$\left[C\right]=3\times 2,\left[B\right]=2\times 3$

Therefore, $\left[C\right]\left[B\right]=3\times 3$ matrix.

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

$=\left[\begin{array}{ccc}2\times 2+0\times 1& 2\times -1+0\times 0& 2\times 0+0\times 1\\ 1\times 2+2\times 1& 1\times -1+2\times 0& 1\times 0+2\times 1\\ -1\times 2+1\times 1& -1\times -1+1\times 0& -1\times 0+1\times 1\end{array}\right]$

$=\left[\begin{array}{ccc}4+0& -2+0& 0+0\\ 2+2& -1+0& 0+2\\ -2+1& 1+0& 0+1\end{array}\right]$

$=\left[\begin{array}{ccc}4& -2& 0\\ 4& -1& 2\\ -1& 1& 1\end{array}\right]$

$3D=3\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 1& -1\end{array}\right]$

$=\left[\begin{array}{ccc}3\times 0& 3\times 1& 3\times 0\\ 3\times -1& 3\times 0& 3\times 0\\ 3\times 0& 3\times 1& 3\times -1\end{array}\right]$

$=\left[\begin{array}{ccc}0& 3& 0\\ -3& 0& 0\\ 0& 3& -3\end{array}\right]$

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

$=\left[\begin{array}{ccc}4-0& -2-3& 0-0\\ 4-\left(-3\right)& -1-0& 2-0\\ -1-0& 1-3& 1-\left(-3\right)\end{array}\right]$

$=\left[\begin{array}{ccc}4& -5& 0\\ 7& -1& 2\\ -1& -2& 4\end{array}\right]$

Final statement:

The matrix operation of $\left[C\right]\left[B\right]-3\left[D\right]=\left[\begin{array}{ccc}4& -5& 0\\ 7& -1& 2\\ -1& -2& 4\end{array}\right]$