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Mathematics Question on Determinants

Find the inverse of each of the matrices(if it exists). [123\024\005]\begin{bmatrix}1&2&3\\\0&2&4\\\0&0&5\end{bmatrix}

Answer

Let A=[123\024\005]\begin{bmatrix}1&2&3\\\0&2&4\\\0&0&5\end{bmatrix}

We have, IAI=1(10-0)-2(0-0)+3(0-0)=10
A11=10-0=10, A12=-(0-0)=0,A13=0-0=0
A21=-(10-0)=-10, A22=5-0=5, A23=-(0-0)=0
A31=8-6=2 ,A32=-(4-0)=-4,A33=2-0=2

adj A=[10−102\05−4\002]\begin{bmatrix}10&-10&2\\\0&5&-4\\\0&0&2\end{bmatrix}

so A-1=1∣A∣\frac{1}{\mid A \mid}adj A=110[10−102\05−4\002]\frac{1}{10}\begin{bmatrix}10&-10&2\\\0&5&-4\\\0&0&2\end{bmatrix}