suppose I have the 4x4 matrix: | x11 x12 x13 x14| |x21 x22 x23 x24| |x31 x32 x33

Question

suppose I have the 4x4 matrix:

| x11 x12 x13 x14|

|x21 x22 x23 x24|

|x31 x32 x33 x34|

|x41 x42 x43 x44 |

how can I find the determinant of it by using the property:

a) If the 1st row of matrix A is the sum of the 1st row of matrix B and C, then we have det(A) = det(B) + det(C)

note: you shall get 4! = 24 term with each team containing four entries. You should have 12 positive terms and 12 negative terms. DO NOT use the cofactor expansion to calculate this!

PS: I know the expansion of an 3x3 matrix is : (x11x22x33)-(x11x23x32)-(x12x21x33)+(x12x23x31)+(x13x23x32)-(x13x22x31), which is writen on my book wothout an proof. So how can I proof the 3x3 also with the 4x4? and I cant use cofactor theorem for this.

Explanation / Answer

A:=matrix([[a11, a12, a13], [b21+c21,b22+c22, b23+c23], [a31,a32,a33]]);

You see that the second row of this matrix is the sum of two row vectors. Now let us define matrices B and C:


> B:=matrix([[a11,a12,a13],[b21,b22,b23],[a31,a32,a33]]);

> C:=matrix([[a11,a12,a13],[c21,c22,c23],[a31,a32,a33]]);

Find the determinant of C, A, B:


> da:=det(A); db:=det(B); dc:=det(C);

da := a11 a33 b22 + a11 a33 c22 - a11 a32 b23 - a11 a32 c23 - b21 a12 a33 + b21 a13 a32 - c21 a12 a33 + c21 a13 a32 + a31 a12 b23 + a31 a12 c23 - a31 a13 b22 - a31 a13 c22 db := a11 a33 b22 - a11 a32 b23 - b21 a12 a33 + b21 a13a32 + a31 a12 b23 - a31 a13 b22 dc := a11 a33 c22 - a11 a32 c23 - c21 a12 a33 + c21 a13 a32 + a31 a12 c23 - a31 a13 c22 You can now see that each term occurring in det(A) or det(B) occurs also in det(A) and vice versa. So indeed det(C)=det(A)+det(B).

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.