Question
Mathematics
Evaluate:
$|\begin{array}{rrr}2& -1& 2\\ \\ 1& 3& 4\\ \\ 1& 2& 1\end{array}|$|[2,-1,2],[],[1,3,4],[],[1,2,1]|
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Step 1 - We can use the Laplace expansion method to evaluate the determinant of the given matrix. Let’s expand along the first row.
$|\begin{array}{rrr}2& -1& 2\\ \\ 1& 3& 4\\ \\ 1& 2& 1\end{array}|=2|\begin{array}{rr}3& 4\\ \\ 2& 1\end{array}|-\left(-1\right)|\begin{array}{rr}1& 4\\ \\ 2& 1\end{array}|+2|\begin{array}{rr}1& 3\\ \\ 2& 1\end{array}|$|[2,-1,2],[],[1,3,4],[],[1,2,1]|=2|[3,4],[],[2,1]|-(-1)|[1,4],[],[2,1]|+2|[1,3],[],[2,1]|
Step 2 - Now, we need to evaluate the determinants of the 2x2 matrices.
$|\begin{array}{rr}3& 4\\ \\ 2& 1\end{array}|=\left(3×1\right)-\left(4×2\right)=-5$|[3,4],[],[2,1]|=(3xx1)-(4xx2)=-5
$|\begin{array}{rr}1& 4\\ \\ 2& 1\end{array}|=\left(1×1\right)-\left(4×2\right)=-7$|[1,4],[],[2,1]|=(1xx1)-(4xx2)=-7
$|\begin{array}{rr}1& 3\\ \\ 2& 1\end{array}|=\left(1×1\right)-\left(3×2\right)=-5$|[1,3],[],[2,1]|=(1xx1)-(3xx2)=-5
Substituting these values in the Laplace expansion formula, we get:
$|\begin{array}{rrr}2& -1& 2\\ \\ 1& 3& 4\\ \\ 1& 2& 1\end{array}|=2\left(-5\right)-\left(-1\right)\left(-7\right)+2\left(-5\right)=-9$|[2,-1,2],[],[1,3,4],[],[1,2,1]|=2(-5)-(-1)(-7)+2(-5)=-9
Answer - The determinant of the given matrix is -9.
Laplace expansion method for determinant.
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