How do you find the determinant of $((-5, 4, 1), (4, 7, 0), (-3, 4, -1))$ ?

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Answer 1 · 2016-06-13T16:25:48

The determinant of the given matrix is $88.$

If $A$ is a matrix, where $A=[(a_11,a_12,a_13), (a_21,a_22,a_23), (a_31,a_32,a_33)]$ , then we define the determinant of the matrix to be

$|A|=|(a_11,a_12,a_13), (a_21,a_22,a_23), (a_31,a_32,a_33)|$

$implies |A|=a_11|(a_22,a_23), (a_32,a_33)|-a_12|(a_21,a_23), (a_31,a_33)|+a_13|(a_21,a_22), (a_31,a_32)|$

$implies |A|=a_11(a_22a_33-a_32a_23)-a_12(a_21a_33-a_31a_23)+a_13(a_21a_32-a_31a_22)$

And then simply this expression.

Now, let $A=[(-5,4,1), (4,7,0), (-3,4,-1)]$

$implies |A|=|(-5,4,1), (4,7,0), (-3,4,-1)|=-5|(7,0), (4,-1)|-4|(4,0), (-3,-1)|+1|(4,7), (-3,4)|$

$implies |A|=-5{7*(-1)-4*0}-4{4(-1)-(-3)*0}+1{4*4-(-3)*7}$

$implies |A|=-5(-7-0)-4(-4-0)+(16+21)$

$implies |A|=35+16+37=88$

Hence, determinant of the given matrix is $88.$

How do you find the determinant of ((-5, 4, 1), (4, 7, 0), (-3, 4, -1))((-5, 4, 1), (4, 7, 0), (-3, 4, -1))?