Question

Find the value of the determinant using the method of expansion by minors; expand on the third row

Answer 1

For the matrix

`\begin{bmatrix}{a_(11)} & {a_(12)} & {a_(13)} \\ {a_(21)} & {a_(22)} & {a_(23)} \\ {a_(31)} & {a_(32)} & {a_(33)}\end{bmatrix}`

the determinant using the method of expansion by minors, expanding on the third row is:

Answer:

First, we compute the determinants of the minors:

`\begin{gathered} \det \begin{bmatrix}{0} & {4} & {} \\ {-1} & {3} & {} \\ {} & {} & {}\end{bmatrix}=0+4=4, \\ \det \begin{bmatrix}{1} & {4} & {} \\ {1} & {3} & {} \\ {} & {} & {}\end{bmatrix}=3-4=-1, \\ \det \begin{bmatrix}{1} & {0} & {} \\ {1} & {-1} & {} \\ {} & {} & {}\end{bmatrix}=-1-0=-1. \end{gathered}`

Therefore:

`\det \begin{bmatrix}{1} & {0} & {4} \\ {1} & {-1} & {3} \\ {0} & {5} & {-2}\end{bmatrix}=0*4-5*(-1)+(-2)*(-1)=5+2=7.`