Two by two matrix examples

Formula: [latex]\det\left(\begin{array}{c}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right)=a_{11}a_{22}-a_{21}a_{12}[/latex].

1. Find the determinant of the matrix [latex]A=\left(\begin{array}{c}-3&1\\-4&2\end{array}\right)[/latex]. [latex]\det(A)=-3(2)-(-4)(1)=-6-(-4)=-2[/latex].
2. Find the determinant of the matrix [latex]B=\left(\begin{array}{c}-1&2\\4&2\end{array}\right)[/latex]. [latex]\det(B)=-1(2)-4(2)=-2-8=-10[/latex].
3. Find the determinant of the matrix [latex]C=\left(\begin{array}{c}-1&2\\2&-4\end{array}\right)[/latex]. [latex]\det(C)=-1(-4)-2(2)=4-4=0[/latex].

Three by three matrix examples

Formula: [latex]\det\left(\begin{array}{c}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right)[/latex][latex]=a_{11}(a_{22}a_{33}-a_{32}a_{23})-a_{12}(a_{21}a_{33}-a_{31}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})[/latex].

This formula is based on expanding along the first row, but the calculation can be done in practice by expanding along any row or column (although the “plus/minus/plus” pattern changes to “minus/plus/minus” if expanding along the second row or column). Tip: expand along the row or column with the most zeros.

1. Find the determinant of the matrix [latex]D=\left(\begin{array}{c}2&2&4\\1&4&-4\\4&4&1\end{array}\right)[/latex].
   [latex]\det(D)=2(4(1)-4(-4))-2(1(1)-4(-4))+4(1(4)-4(4))[/latex][latex]=2(20)-2(17)+4(-12)=40-34-48=-42[/latex].
2. Find the determinant of the matrix [latex]E=\left(\begin{array}{c}2&2&4\\1&4&-4\\5&8&4\end{array}\right)[/latex].
   [latex]\det(E)=2(4(4)-8(-4))-2(1(4)-5(-4))+4(1(8)-5(4))[/latex][latex]=2(48)-2(24)+4(-12)=96-48-48=0[/latex].
3. Find the determinant of the matrix [latex]F=\left(\begin{array}{c}-4&2&3\\2&3&0\\0&2&-1\end{array}\right)[/latex].
   Expand along the second row.
   [latex]\det(F)=-2(2(-1)-2(3))+3(-4(-1)-0(3))+0[/latex][latex]=-2(-8)+3(4)=16+12=28[/latex].