Q10. - Algebra 2

Question

 Use matrices $A$, B, and C to find the following.

$A=\left[\begin{array}{cc}2& 3\\ 5& 6\end{array}\right],B=\left[\begin{array}{cc}-1& 7\\ 0& 4\end{array}\right],C=\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]$

10. $4A+2B-C$

Step-by-Step Solution

Answer

The final matrix after performing the indicated matrix operations is

$4A+2B-C=\left[\begin{array}{cc}-3& 30\\ 26& 27\\ & \end{array}\right]$.

Step 1 - Substitute the matrices

Substitute $\left[\begin{array}{cc}2& 3\\ 5& 6\\ & \end{array}\right]$ for A, $\left[\begin{array}{cc}-1& 7\\ 0& 4\\ & \end{array}\right]$ for B and $\left[\begin{array}{cc}9& -4\\ -6& 5\\ & \end{array}\right]$ for C into $4A+2B-C$.

 

$4A+2B-C=4\left[\begin{array}{cc}2& 3\\ 5& 6\\ & \end{array}\right]+2\left[\begin{array}{cc}-1& 7\\ 0& 4\\ & \end{array}\right]-\left[\begin{array}{cc}9& -4\\ -6& 5\\ & \end{array}\right]$

Step 2 - Perform the scalar multiplication

Scalar multiplication - The product of a scalar k and an $m\times n$ matrix is an $m\times n$ matrix in which each element equals k times the corresponding elements of the original matrix.

Multiply each element of the first matrix $\left[\begin{array}{cc}2& 3\\ 5& 6\\ & \end{array}\right]$ by 4 and second matrix $\left[\begin{array}{cc}-1& 7\\ 0& 4\\ & \end{array}\right]$ by 2.

$\begin{array}{c}4A+2B-C=4\left[\begin{array}{cc}2& 3\\ 5& 6\end{array}\right]+2\left[\begin{array}{cc}-1& 7\\ 0& 4\end{array}\right]-\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]\\ =\left[\begin{array}{cc}4\left(2\right)& 4\left(3\right)\\ 4\left(5\right)& 4\left(6\right)\end{array}\right]+\left[\begin{array}{cc}2\left(-1\right)& 2\left(7\right)\\ 2\left(0\right)& 2\left(4\right)\end{array}\right]-\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]\\ =\left[\begin{array}{cc}8& 12\\ 20& 24\end{array}\right]+\left[\begin{array}{cc}-2& 14\\ 0& 8\end{array}\right]-\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]\end{array}$

Step 3 - Perform the matrix addition/subtraction

Matrix addition/subtraction - If A and B are two matrices of order $m\times n$, then the addition/subtraction of the two matrices will also yield a matrix of order $m\times n$wherein each element is the sum/difference of the corresponding elements.

$\begin{array}{c}4A+2B-C=\left[\begin{array}{cc}8& 12\\ 20& 24\end{array}\right]+\left[\begin{array}{cc}-2& 14\\ 0& 8\end{array}\right]-\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]\\ =\left[\begin{array}{cc}8-2-9& 12+14-\left(-4\right)\\ 20+0-\left(-6\right)& 24+8-5\end{array}\right]\\ =\left[\begin{array}{cc}-3& 30\\ 26& 27\end{array}\right]\end{array}$