Q. 49MYS - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible.

$2\left[\begin{array}{ccc}4& 9& -8\\ 6& -11& -2\\ 12& -10& 3\\ & & \end{array}\right]+3\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 5\\ & & \end{array}\right]$

Step-by-Step Solution

Answer

After performing matrix operations, resulting matrix is $\left[\begin{array}{ccc}11& 24& -7\\ 18& -13& 8\\ 33& -8& 21\\ & & \end{array}\right]$

Step 1. Write operation with matrices

Let A and B be two $m\times n$ matrices, then following operations are defined:

• Matrix Addition- $A+B$ is an $m\times n$ matrix in which each element is the sum of the corresponding elements of A and B
•  Matrix Subtraction- $A-B$ is an $m\times n$ matrix in which each element is the difference of the corresponding elements of A and B
• Scalar Multiplication- The product of a scalar $k$ with $m\times n$ matrix is an $m\times n$ matrix in which each element equals k times the corresponding elements of the original matrix.

Step 2. Perform Scalar Multiplication

First matrix operation applied in given expression is scalar multiplication. Multiply given two matrices with scalars as shown below

$2\left[\begin{array}{ccc}4& 9& -8\\ 6& -11& -2\\ 12& -10& 3\\ & & \end{array}\right]+3\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 5\\ & & \end{array}\right]=\left[\begin{array}{ccc}8& 18& -16\\ 12& -22& -4\\ 24& -20& 6\\ & & \end{array}\right]+\left[\begin{array}{ccc}3& 6& 9\\ 6& 9& 12\\ 9& 12& 15\\ & & \end{array}\right]$

Step 3. Perform Matrix Addition

As both the matrices are of same dimension, so apply matrix addition to get,

$\begin{array}{c}\left[\begin{array}{ccc}8& 18& -16\\ 12& -22& -4\\ 24& -20& 6\end{array}\right]+\left[\begin{array}{ccc}3& 6& 9\\ 6& 9& 12\\ 9& 12& 15\end{array}\right]=\left[\begin{array}{ccc}8+3& 18+6& -16+9\\ 12+6& -22+9& -4+12\\ 24+9& -20+12& 6+15\end{array}\right]\\ =\left[\begin{array}{ccc}11& 24& -7\\ 18& -13& 8\\ 33& -8& 21\end{array}\right]\end{array}$