Question

The matrix A is defined as $A=\left[\begin{array}{ccc}1& 2& -3\\ 0& 3& 2\\ 0& 0& -2\end{array}\right]$ the determinant of matrix $B=3A^3+5A^2-6A+2I$ is

• A. 110
• B. 1100
• C. 4400
• D. 440

Answer 1

The correct option is C 4400

$|A-\lambda I|=0$

$\left|\begin{array}{ccc}1-\lambda & 2& -3\\ 0& 3-\lambda & 2\\ 0& 0& -2-\lambda \end{array}\right|=0$

$(1-\lambda )(3-\lambda )(-2-\lambda )=0$
$\lambda =1,3,-2$

Eigen values of $A^3=1,27,-8$
Eigen values of $A^2=1,9,4$
Eigen values of $A=1,3,-2$

Eigen values of I = 1,1,1
Eigen values of matrix B,
First eigen values $=3(1{)}^3+5(1{)}^2-6\left(1\right)+2\left(1\right)=4$
Second eigen value $=3\left(27\right)+5\left(9\right)-6\left(3\right)+2\left(1\right)=110$
Thir eigen value $=3(-8)+5\left(4\right)-6(-2)+2\left(1\right)=10$
Hence, required eigen values are, 4, 110 and 10
Determinant of matrix $B={\lambda }_1{\lambda }_2{\lambda }_3=4400$