Q.If $A=\begin{pmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{pmatrix}$ and $A^3-6A^2+7A+kI=O$, find the value of $k$.

Step 1. Compute $A^2=A\cdot A$.

Row $1\cdot$Col $1=1(1)+0(0)+2(2)=5$;\ Row $1\cdot$Col $2=1(0)+0(2)+2(0)=0$;\ Row $1\cdot$Col $3=1(2)+0(1)+2(3)=8$

Row $2\cdot$Col $1=0(1)+2(0)+1(2)=2$;\ Row $2\cdot$Col $2=0(0)+2(2)+1(0)=4$;\ Row $2\cdot$Col $3=0(2)+2(1)+1(3)=5$

Row $3\cdot$Col $1=2(1)+0(0)+3(2)=8$;\ Row $3\cdot$Col $2=2(0)+0(2)+3(0)=0$;\ Row $3\cdot$Col $3=2(2)+0(1)+3(3)=13$

Step 2. Compute $A^3=A^2\cdot A$.

Row $1$: $5(1)+0(0)+8(2)=21;\ 5(0)+0(2)+8(0)=0;\ 5(2)+0(1)+8(3)=34$

Row $2$: $2(1)+4(0)+5(2)=12;\ 2(0)+4(2)+5(0)=8;\ 2(2)+4(1)+5(3)=23$

Row $3$: $8(1)+0(0)+13(2)=34;\ 8(0)+0(2)+13(0)=0;\ 8(2)+0(1)+13(3)=55$

Step 3. Form $A^3-6A^2+7A$. …

• Arithmetic slip in one of the nine entries of $A^2$ or $A^3$, which throws off every later step …