Answer to Question #236802 in Algebra for moe

Question

You are given a matrix 

$A=\left(\begin{array}{cccc}1 & -2 & -1\\ 2 & -3 & 0 \\1& -1 & -3\end{array} \right).$

Which of the following are true about A?

 

1. $det (A)=-4$
2. $adj (A)=\left(\begin{array}{cccc}9 & -5 & -3\\ 6 & -2 & -2 \\1& -1 & 1\end{array} \right)$
3. $A^{-1}=\left(\begin{array}{cccc}-\frac{9}{4} & \frac{5}{4} & \frac{3}{4}\\ -\frac{3}{2} & \frac{1}{2} & \frac{1}{2} \\-\frac{1}{4}& \frac{1}{4} & -\frac{1}{4}\end{array} \right)$
4. The system AX=0 has no solution, where 0 denotes a zero matrix of order 3×1.
5. $adj (A)=\left(\begin{array}{cccc}9 & -5 & -3\\ 6 & -2 & -2 \\1& -1 & 1\end{array} \right)^{T}$

Answer 1

$det(A)=−4$

$A^{-1}=\left(\begin{array}{cccc}-\frac{9}{4} & \frac{5}{4} & \frac{3}{4}\\ -\frac{3}{2} & \frac{1}{2} & \frac{1}{2} \\-\frac{1}{4}& \frac{1}{4} & -\frac{1}{4}\end{array} \right)$