# (i) find solution

Question:

(i) $\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$

(ii) $\left|\begin{array}{rrr}1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2\end{array}\right|$

Solution:

(i) The given determinant is $\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$.

By the definition of minors and cofactors, we have:

$M_{11}=$ minor of $a_{11}=\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|=1$

$M_{12}=$ minor of $a_{12}=\left|\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right|=0$

$M_{13}=$ minor of $a_{13}=\left|\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right|=0$

$M_{21}=$ minor of $a_{21}=\left|\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right|=0$

$M_{22}=$ minor of $a_{22}=\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|=1$

$M_{23}=$ minor of $a_{23}=\left|\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right|=0$

$M_{31}=$ minor of $a_{31}=\left|\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right|=0$

$\mathrm{M}_{32}=$ minor of $a_{32}=\left|\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right|=0$

$\mathrm{M}_{33}=$ minor of $a_{33}=\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|=1$

$A_{11}=$ cofactor of $a_{11}=(-1)^{1+1} M_{11}=1$

$A_{12}=$ cofactor of $a_{12}=(-1)^{1+2} M_{12}=0$

$A_{13}=$ cofactor of $a_{13}=(-1)^{1+3} M_{13}=0$

$A_{21}=$ cofactor of $a_{21}=(-1)^{2+1} M_{21}=0$

$A_{22}=$ cofactor of $a_{22}=(-1)^{2+2} M_{22}=1$

$A_{23}=$ cofactor of $a_{23}=(-1)^{2+3} M_{23}=0$

$A_{31}=$ cofactor of $a_{31}=(-1)^{3+1} M_{31}=0$

$A_{32}=$ cofactor of $a_{32}=(-1)^{3+2} M_{32}=0$

$A_{33}=$ cofactor of $a_{33}=(-1)^{3+3} M_{33}=1$

(ii) The given determinant is $\left|\begin{array}{rrr}1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2\end{array}\right|$.

By definition of minors and cofactors, we have:

$M_{11}=$ minor of $a_{11}=\left|\begin{array}{cc}5 & -1 \\ 1 & 2\end{array}\right|=10+1=11$

$M_{12}=$ minor of $a_{12}=\left|\begin{array}{cc}3 & -1 \\ 0 & 2\end{array}\right|=6-0=6$

$M_{13}=$ minor of $a_{13}=\left|\begin{array}{cc}3 & 5 \\ 0 & 1\end{array}\right|=3-0=3$

$M_{21}=$ minor of $a_{21}=\left|\begin{array}{cc}0 & 4 \\ 1 & 2\end{array}\right|=0-4=-4$

$M_{22}=$ minor of $a_{22}=\left|\begin{array}{cc}1 & 4 \\ 0 & 2\end{array}\right|=2-0=2$

$M_{23}=$ minor of $a_{23}=\left|\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right|=1-0=1$

$M_{31}=$ minor of $a_{31}=\left|\begin{array}{cc}0 & 4 \\ 5 & -1\end{array}\right|=0-20=-20$

$M_{33}=$ minor of $a_{33}=\left|\begin{array}{cc}1 & 0 \\ 3 & 5\end{array}\right|=5-0=5$

$M_{32}=$ minor of $a_{32}=\left|\begin{array}{cc}1 & 4 \\ 3 & -1\end{array}\right|=-1-12=-13$

$\mathrm{M}_{33}=$ minor of $\mathrm{a}_{33}=\left|\begin{array}{ll}1 & 0 \\ 3 & 5\end{array}\right|=5-0=5$

$\mathrm{A}_{11}=$ cofactor of $a_{11}=(-1)^{1+1} \mathrm{M}_{11}=11$

$\mathrm{~A}_{12}=$ cofactor of $a_{12}=(-1)^{1+2} \mathrm{M}_{12}=-6$

$\mathrm{~A}_{13}=$ cofactor of $a_{13}=(-1)^{1+3} \mathrm{M}_{13}=3$

$\mathrm{~A}_{21}=$ cofactor of $a_{21}=(-1)^{2+1} \mathrm{M}_{21}=4$

$\mathrm{~A}_{22}=$ cofactor of $a_{22}=(-1)^{2+2} \mathrm{M}_{22}=2$

$\mathrm{~A}_{23}=$ cofactor of $a_{23}=(-1)^{2+3} \mathrm{M}_{23}=-1$

$\mathrm{~A}_{31}=$ cofactor of $a_{31}=(-1)^{3+1} \mathrm{M}_{31}=-20$

$\mathrm{~A}_{32}=$ cofactor of $a_{32}=(-1)^{3+2} \mathrm{M}_{32}=13$

$\mathrm{~A}_{33}=$ cofactor of $a_{33}=(-1)^{3+3} \mathrm{M}_{33}=5$