find which of the given matrices are nonsingular. a) begin{bmatrix}1 & 2 &-3 -1 & 2&3 0 &8&0 end{bmatrix} b)begin{bmatrix}1 & 2 &-3 -1 & 2&3 0 &1&1 end{bmatrix} c) begin{bmatrix}1 & 1 &2 -1 & 3&4 -5 &7&8 end{bmatrix} d) begin{bmatrix}1 & 1 &4&-1 1 & 2&3&2 -1 &3&2&1-2&6&12&-4 end{bmatrix}

find which of the given matrices are nonsingular.
a) $\left[\begin{array}{ccc}1& 2& -3\\ -1& 2& 3\\ 0& 8& 0\end{array}\right]$
b)$\left[\begin{array}{ccc}1& 2& -3\\ -1& 2& 3\\ 0& 1& 1\end{array}\right]$
c) $\left[\begin{array}{ccc}1& 1& 2\\ -1& 3& 4\\ -5& 7& 8\end{array}\right]$
d) $\left[\begin{array}{cccc}1& 1& 4& -1\\ 1& 2& 3& 2\\ -1& 3& 2& 1\\ -2& 6& 12& -4\end{array}\right]$
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averes8
Step 1
Since you have posted a multiple sub-parts problems ,we solve first 3 sub-parts for you .To get the remaining sub-parts solved please repost the complete question ,and mention sub-parts to be solved.
We have given some matrices and we have to determine which matrices are non-singular among them.
Step 2 First note that a matrix A is said to be nonsingular if $det\left(A\right)\ne 0$.We will use this to determine the non singularity of the matrices .
a) we have $A=\left[\begin{array}{ccc}1& 2& -3\\ -1& 2& 3\\ 0& 8& 0\end{array}\right]$
Now $det\left(A\right)=1×\left(0-24\right)-2×\left(0-0\right)-3×\left(-8-0\right)=-24+24=0$
Since determinant is 0 so the matrix is singular ,not non-singular.
b) $B=\left[\begin{array}{ccc}1& 2& -3\\ -1& 2& 3\\ 0& 1& 1\end{array}\right]$
Now $det\left(B\right)=1×\left(2-3\right)-2×\left(-1-0\right)-3×\left(-1-0\right)=-1+2+3=4\ne 0$
Sine the determinant is non-zero so the matrix is non-singular .
c) $C=\left[\begin{array}{ccc}1& 1& 2\\ -1& 3& 4\\ -5& 7& 8\end{array}\right]$
Now, $det\left(C\right)=1×\left(24-28\right)-1×\left(-8+20\right)+2×\left(-7+15\right)=0$
Since determinant is 0 so the matrix is singular ,not non-singular.
Jeffrey Jordon