Compute the determinant of the matrix by cofactor...

Compute the determinant of the matrix by cofactor expansion. \begin{bmatrix} 2 & 2 & -2 & 5 & -4 \\ 0 & 3 & 2 & 1 & 0 \\ 0 & 0 & -2 & 3 & 7 \\ 0 & 0 & 0 & -1 & -5 \\ 0 & 0 & 0 & 0 & 2 \end{bmatrix}

Transcript

00:01 Hello students, we are given a matrix and we need to find a determinant of matrix by expansion of cofactors, right? so what would be my cofactors? cofactors are decided by c .i .j, which is equal to minus i -rish -to -power, i plus j and m -i -j, right? where i and j represent number of rows and number of columns.

00:21 Here, number of rows and number of columns are 4 -cross 4.

00:24 So maximum value of i can be 4 and j also can be 4, right? if i talk about c11, c11 would be, so c11 would be minus 1 raise to power 1 plus 1 which is even means minus 1 raise to power, e1 values always plus 1 and minus 1 raise to power odd values minus 1 always.

00:47 So plus 1 will come and we need to calculate m11 which is in respect to this matrix right? so we need to calculate a determinant for this of this matrix for calculating m11.

01:00 So this would be 6 multiply by minus 6 and 4 into 2 8 and minus 4 multiply by 2 2 minus 4 and at the end 12 multiply by minus 4 plus 3 right so my final answer would be 8 minus 6 2 2 into 6 is 12 and this is 0 minus 0 and minus 1 and minus 12 and that can be calculated as 0 so m11 value would be 0 so this multiply by 0 then c11 is 0.

01:31 Right.

01:32 Then i can calculate, i need to calculate c12 again then this would be minus 1 and minus 1 and m12 value.

01:40 M1 2 value can be calculated with respect to 3.

01:43 So we can calculate what is my m1 2 value? m1 2 value would be m1 2.

01:50 This can be calculated by using 4 multiplied by minus 6 and plus 8.

01:55 4 multiply by minus 6 plus 8.

01:59 Then other terms are 0 multiplied and 0 multiplied.

02:02 So this will become 0, right? so only this value will come 8 minus 6 is 2.

02:08 Then this would become 8.

02:10 So m1 2 is 8.

02:12 So it can be written as 8.

02:14 So minus 8 is the answer.

02:16 Now let us talk about c13 value.

02:20 So if i talk about c13 value, so let's assume this value to be a because it will be cancel out become zero in matrix equation, right? in matrix equation or in expansion by cofactors, right? in calculating determinant.

02:37 So if i need now, we are going to calculate c14.

02:42 So for c14 i can say is c1414 can be calculated by minus minus.

02:51 1, raise to power of 4 plus 1, which is minus 1 again and multiply by m14.

02:59 M14 can be calculated by in respect to this element...

Question

Compute the determinant of the matrix by cofactor expansion. \begin{bmatrix} 2 & 2 & -2 & 5 & -4 \\ 0 & 3 & 2 & 1 & 0 \\ 0 & 0 & -2 & 3 & 7 \\ 0 & 0 & 0 & -1 & -5 \\ 0 & 0 & 0 & 0 & 2 \end{bmatrix}

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