Chapter 8.4, Problem 41E

i.

To determine

To calculate: Find the determinant of the given matrix. Expand by cofactors using Row 2

  $\left[\begin{array}{ccc}5& 0& -3\\ 0& 12& 4\\ 1& 6& 3\end{array}\right]$

Answer to Problem 41E

  $\det \left(A\right)=96$

Explanation of Solution

Concept used:

Consider a $3\times 3$ matrix A, such that $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$

Minor of $a_{11}=M_{11}=\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|$

Minor of $a_{12}=M_{12}=\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|$

Minor of $a_{13}=M_{13}=\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

Minor of $a_{21}=M_{21}=\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|$

Minor of $a_{22}=M_{22}=\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{31}& a_{33}\end{array}\right|$

Minor of $a_{23}=M_{23}=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{31}& a_{32}\end{array}\right|$

Minor of $a_{31}=M_{31}=\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{22}& a_{23}\end{array}\right|$

Minor of $a_{32}=M_{32}=\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{21}& a_{23}\end{array}\right|$

Minor of $a_{33}=M_{33}=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$

Cofactor of $a_{ij}=C_{ij}={\left(-1\right)}^{i+j}{M}_{ij}$

Also iIf A is any square matrix, then the determinant of A expanding along i^throw in terms of cofactor is given by

  $\left|A\right|=a_{i1}{C}_{i1}+a_{i2}{C}_{i2}+\cdots +a_{in}{C}_{in}$

For $n=3,i=2$

  $\left|A\right|=a_{21}{C}_{21}+a_{22}{C}_{22}+a_{23}{C}_{23}$

Calculation:

Let $A=\left[\begin{array}{ccc}5& 0& -3\\ 0& 12& 4\\ 1& 6& 3\end{array}\right]$

Expand above matrix along row 2 using cofactor

  $\begin{array}{l}\left|A\right|=a_{21}{C}_{21}+a_{22}{C}_{22}+a_{23}{C}_{23}\\ \left|A\right|=-0\left|\begin{array}{cc}0& -3\\ 6& 3\end{array}\right|+12\left|\begin{array}{cc}5& -3\\ 1& 3\end{array}\right|-4\left|\begin{array}{cc}5& 0\\ 1& 6\end{array}\right|\\ \left|A\right|=0+12\left(15+3\right)-4\left(30-0\right)\\ \left|A\right|=96\end{array}$

ii.

To determine

To calculate: Find the determinant of the given matrix. Expand by cofactors using Column 2

  $\left[\begin{array}{ccc}5& 0& -3\\ 0& 12& 4\\ 1& 6& 3\end{array}\right]$

Answer to Problem 41E

  $\det \left(A\right)=96$

Explanation of Solution

Concept used:

Consider a $3\times 3$ matrix A, such that $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$

Minor of $a_{11}=M_{11}=\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|$

Minor of $a_{12}=M_{12}=\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|$

Minor of $a_{13}=M_{13}=\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

Minor of $a_{21}=M_{21}=\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|$

Minor of $a_{22}=M_{22}=\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{31}& a_{33}\end{array}\right|$

Minor of $a_{23}=M_{23}=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{31}& a_{32}\end{array}\right|$

Minor of $a_{31}=M_{31}=\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{22}& a_{23}\end{array}\right|$

Minor of $a_{32}=M_{32}=\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{21}& a_{23}\end{array}\right|$

Minor of $a_{33}=M_{33}=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$

Cofactor of $a_{ij}=C_{ij}={\left(-1\right)}^{i+j}{M}_{ij}$

Also if A is any square matrix, then the determinant of A expanding along j^thcolumn in terms of cofactor is given by

  $\left|A\right|=a_{1j}{C}_{1j}+a_{2j}{C}_{2j}+\cdots +a_{nj}{C}_{nj}$

For $n=3,j=2$

  $\left|A\right|=a_{12}{C}_{12}+a_{22}{C}_{22}+a_{32}{C}_{32}$

Calculation:

Let $A=\left[\begin{array}{ccc}5& 0& -3\\ 0& 12& 4\\ 1& 6& 3\end{array}\right]$

Expand above matrix along column 2 using cofactor

  $\begin{array}{l}\left|A\right|=a_{12}{C}_{12}+a_{22}{C}_{22}+a_{32}{C}_{32}\\ \left|A\right|=-0\left|\begin{array}{cc}0& 4\\ 1& 3\end{array}\right|+12\left|\begin{array}{cc}5& -3\\ 1& 3\end{array}\right|-6\left|\begin{array}{cc}5& -3\\ 0& 4\end{array}\right|\\ \left|A\right|=0+12\left(15+3\right)-6\left(20-0\right)\\ \left|A\right|=96\end{array}$