Find the inverse of a 3x3 matrix

[frctl]

attempt 1 finding determinant
|A| = (4 + 6 + 0) - (15 + 0 + 32)
|A| = -37

attempt 2 finding determinant
|A| = 1(5 x 8 - 3 x 0) - -2(2 x 8 - 3 x 1) + 3(2 x 0 - 5 x 1)
|A| = 40 + 26 + -15
|A| = 51

Which method have I been successful at finding the determinant?

 

[Dr.Peterson]

Just check your work at each step.

In the first, you dropped a digit in one of the six products.

In the second, you have a double negative (that is, you both subtracted and changed a sign).

If you fix each of these errors, you will get the same result both ways. As they are, both are wrong.

One of the keys to doing math well is to learn to work slowly, check each step, and then check the final result when possible. Trying two ways is one way to check the result, but takes more work than the first two practices.

 

[frctl]

Thank you Dr.Peterson, I have fixed my second attempt and obtained a determinant of -1.

 

[Dr.Peterson]

Good. And the error in the first attempt was writing 4 instead of 40. You know both methods, and just have to be careful ...

 

[frctl]

matrix of minors
[5 3] [2 3] [2 5]
0 8 1 8 1 0
[2 3] [1 3] [1 2]
0 8 [1 8] [10 ]
[2 3] [1 3] [1 2]
5 3 [2 3] [2 5]

cofactor matrix
+40 -13 +5
-16 +5 -2
+9 -3 +1

reflected
40 -16 9
-13 5 3
5 -2 +1

times the determinant
-40 16 -9
13 -5 -3
-5 2 -1

I comparison with the answer I have sign errors
The answer at the back of the book is:
-40 16 9
13 -5 -3
5 -2 -1

 

[Dr.Peterson]

Look at each entry that has the wrong sign, and trace through the steps that led to it. You should be able to find the errors easily. They are, of course, sign errors ... all of them, I believe, in the first step (cofactor matrix).

And next time, do the same thing, but before looking in the back of the book, as practice for exams.