Consider the Matrix M (below). Find the determiannt of the matrix M by using; (a) cofactor expansion along the first row, (b) cofactor expansion along the second column

Matrix M is

-2 1 -4

1 -1  5

3  0  2

Let A be a square nxn matrix. Then, for each entry Aij (where 1 </= i < n, 1 </= j < n), the minor of Aij is A(^i;^j), the determinant of the (n - 1) (n - 1) matrix A(^i;^j). Then the cofactor of Aij is the minor of Aij multiplied by (-1)^(i+j) , i.e. multiplied by +1 if the sum of the row and column we are considering is even and by -1 if this sum is odd.

Of which the formula is n(Sigma)i=1 (-1)^(i+j)| A(^i;^j)|

a) Using cofactor expansion along the first row we have that the determinant of the matrix M is

(-2) |-1(2)+ 5(0)| + (-1) |1(2) + (3)(5)| +(-4)|1(0) + (3)(-1)|

(-2)(-2) - (1)(-13) + (-4)(3)

= 5

b) Using cofactor expansion along the second row we have that the determinant of the matrix M is

-(1) |1(2)+ 5(3)| + (-1) |(-2)(2) + (-4)(3)| -(0)|(-2)(5) + (1)(-4)|

-(1)(-13) + (-1)(-8) - (0)(6)

= 5