 # simplify the following matrix expression: begin{bmatrix}1 & 5 2 & -3-3&7 end{bmatrix}-2begin{bmatrix}2 & -3 8 & 5-1&-1 end{bmatrix}=? ruigE 2021-02-08 Answered
simplify the following matrix expression:
$\left[\begin{array}{cc}1& 5\\ 2& -3\\ -3& 7\end{array}\right]-2\left[\begin{array}{cc}2& -3\\ 8& 5\\ -1& -1\end{array}\right]=?$
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Step 1
Given that:
$|\begin{array}{cc}1& 5\\ 2& -3\\ -3& 7\end{array}|-2|\begin{array}{cc}2& -3\\ 8& 5\\ -1& -1\end{array}|$
To find:
We have to simplify the given expression.
First we find:
$2|\begin{array}{cc}2& -3\\ 8& 5\\ -1& -1\end{array}|$
In this multiplication each entry in the matrix is multiplied by the scalar 2.
$⇒2|\begin{array}{cc}2& -3\\ 8& 5\\ -1& -1\end{array}|=|\begin{array}{cc}2\left(2\right)& 2\left(-3\right)\\ 2\left(8\right)& 2\left(5\right)\\ 2\left(-1\right)& 2\left(-1\right)\end{array}|$
$2|\begin{array}{cc}2& -3\\ 8& 5\\ -1& -1\end{array}|=|\begin{array}{cc}4& -6\\ 16& 10\\ -2& -2\end{array}|$
Step 2
$⇒|\begin{array}{cc}1& 5\\ 2& -3\\ -3& 7\end{array}|-2|\begin{array}{cc}2& -3\\ 8& 5\\ -1& -1\end{array}|=|\begin{array}{cc}1& 5\\ 2& -3\\ -3& 7\end{array}|-|\begin{array}{cc}4& -6\\ 16& 10\\ -2& -2\end{array}|$
Now we substracting this two matrices.
Here the order of matrices is same.
So, the difference between this two matrices is obtained by substracting the corresponding elements of the matrices.
We get,
$|\begin{array}{cc}1& 5\\ 2& -3\\ -3& 7\end{array}|-2|\begin{array}{cc}2& -3\\ 8& 5\\ -1& -1\end{array}|=|\begin{array}{cc}1-4& 5-\left(-6\right)\\ 2-16& -3-10\\ -3-\left(-2\right)& 7-\left(-2\right)\end{array}|$
$=|\begin{array}{cc}-3& 5+6\\ -14& -13\\ -3+2& 7+2\end{array}|$
$⇒|\begin{array}{cc}1& 5\\ 2& -3\\ -3& 7\end{array}|-2|\begin{array}{cc}2& -3\\ 8& 5\\ -1& -1\end{array}|=|\begin{array}{cc}-3& 11\\ -14& -13\\ -1& 9\end{array}|$
Hence the solution.
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