Determine , if possible , the value of x which will male the following true: begin{bmatrix}4 & 3 1 & x end{bmatrix}+2begin{bmatrix}2 1 end{bmatrix}begin{bmatrix}3 & x end{bmatrix}=begin{bmatrix}16 & 11 7 & 6 end{bmatrix}

Determine , if possible , the value of x which will male the following true: begin{bmatrix}4 & 3 1 & x end{bmatrix}+2begin{bmatrix}2 1 end{bmatrix}begin{bmatrix}3 & x end{bmatrix}=begin{bmatrix}16 & 11 7 & 6 end{bmatrix}

Question
Matrices
asked 2021-03-18
Determine , if possible , the value of x which will male the following true:
\(\begin{bmatrix}4 & 3 \\1 & x \end{bmatrix}+2\begin{bmatrix}2 \\1 \end{bmatrix}\begin{bmatrix}3 & x \end{bmatrix}=\begin{bmatrix}16 & 11 \\7 & 6 \end{bmatrix}\)

Answers (1)

2021-03-19
Step 1
Product of two matrices:
\(\begin{bmatrix}a\\b \end{bmatrix}\begin{bmatrix}c & d \end{bmatrix}=\begin{bmatrix}ab & ad \\bc & bd \end{bmatrix}\)
Product of a scalar and a matrix:
\(k\begin{bmatrix}a & b \\c & d \end{bmatrix} =\begin{bmatrix}ka & kb \\ kc & kd \end{bmatrix}\)
If two matrices are equal then the corresponding elements of the matrices are equal.
Step 2
Given matrix equation is:
\(\begin{bmatrix}4 & 3 \\1 & x \end{bmatrix}+2\begin{bmatrix}2 \\1 \end{bmatrix}\begin{bmatrix}3 & x \end{bmatrix}=\begin{bmatrix}16 & 11 \\7 & 6 \end{bmatrix}\)
We need to find the value of x.
First, we need to simplify the left-hand side of the equation, and then we have to compare the elements of the left-hand side matrix and the right-hand side matrix.
\(\begin{bmatrix}4 & 3 \\1 & x \end{bmatrix}+2\begin{bmatrix}2\times3&2\times x \\1\times 3&1\times x \end{bmatrix}=\begin{bmatrix}16 & 11 \\7 & 6 \end{bmatrix}\)
\(\begin{bmatrix}4 & 3 \\1 & x \end{bmatrix}+2\begin{bmatrix}6&2x \\3&x \end{bmatrix}=\begin{bmatrix}16 & 11 \\7 & 6 \end{bmatrix}\)
\(\begin{bmatrix}4 & 3 \\1 & x \end{bmatrix}+\begin{bmatrix}2(6)&2(2x) \\2(3)&2(x) \end{bmatrix}=\begin{bmatrix}16 & 11 \\7 & 6 \end{bmatrix}\)
Step 3
Now,
\(\begin{bmatrix}4 & 3 \\1 & x \end{bmatrix}+\begin{bmatrix}12&4x \\6&2x \end{bmatrix}=\begin{bmatrix}16 & 11 \\7 & 6 \end{bmatrix}\)
\(\begin{bmatrix}4+12 & 3+4x \\1+6 & x+2x \end{bmatrix}=\begin{bmatrix}16 & 11 \\7 & 6 \end{bmatrix}\)
\(\begin{bmatrix}16 & 3+4x \\7 & 3x \end{bmatrix}=\begin{bmatrix}16 & 11 \\7 & 6 \end{bmatrix}\)
On comparing both sides, we get
\(3+4x=11 \text{ and } 3x=6\)
\(4x=11-3 \text{ and } x=\frac{6}{3}\)
\(4x=8 \text{ and } x=2\)
\(x=\frac{8}{4} \text{ and } x=2\)
\(x=2 \text{ and } x=2\)
Step 4
Answer: The value of x is 2.
0

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