sodni3

2021-03-18

Determine , if possible , the value of x which will male the following true:
$\left[\begin{array}{cc}4& 3\\ 1& x\end{array}\right]+2\left[\begin{array}{c}2\\ 1\end{array}\right]\left[\begin{array}{cc}3& x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$

Step 1
Product of two matrices:
$\left[\begin{array}{c}a\\ b\end{array}\right]\left[\begin{array}{cc}c& d\end{array}\right]=\left[\begin{array}{cc}ab& ad\\ bc& bd\end{array}\right]$
Product of a scalar and a matrix:
$k\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}ka& kb\\ kc& kd\end{array}\right]$
If two matrices are equal then the corresponding elements of the matrices are equal.
Step 2
Given matrix equation is:
$\left[\begin{array}{cc}4& 3\\ 1& x\end{array}\right]+2\left[\begin{array}{c}2\\ 1\end{array}\right]\left[\begin{array}{cc}3& x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$
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.
$\left[\begin{array}{cc}4& 3\\ 1& x\end{array}\right]+2\left[\begin{array}{cc}2×3& 2×x\\ 1×3& 1×x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$
$\left[\begin{array}{cc}4& 3\\ 1& x\end{array}\right]+2\left[\begin{array}{cc}6& 2x\\ 3& x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$
$\left[\begin{array}{cc}4& 3\\ 1& x\end{array}\right]+\left[\begin{array}{cc}2\left(6\right)& 2\left(2x\right)\\ 2\left(3\right)& 2\left(x\right)\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$
Step 3
Now,
$\left[\begin{array}{cc}4& 3\\ 1& x\end{array}\right]+\left[\begin{array}{cc}12& 4x\\ 6& 2x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$
$\left[\begin{array}{cc}4+12& 3+4x\\ 1+6& x+2x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$
$\left[\begin{array}{cc}16& 3+4x\\ 7& 3x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$
On comparing both sides, we get

Step 4
Answer: The value of x is 2.

Jeffrey Jordon