|2 0 4|

|3 1 5|
|-1 1 -2|

I needed the inverse of the matrix above and is the one below correct? If so how do I put the problem down to make it easier to solve. This took me forever.

|-7/2 2 -2|
|1/2 0 1|
|2 -1 1|

To find the inverse of a matrix, you can use several methods, such as the Gauss-Jordan elimination method or the adjugate method. Since you mentioned that the process took you a long time, let's use the adjugate method, which tends to be simpler for small matrices.

To find the inverse of a 3x3 matrix, follow these steps:

Step 1: Write down the original matrix:

|2 0 4|
|3 1 5|
|-1 1 -2|

Step 2: Find the determinant of the original matrix. The determinant of a 3x3 matrix can be calculated as:

det(A) = (a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31))

For the matrix above, we have:

det(A) = (2((1)(-2) - (1)(5)) - 0((3)(-2) - (1)(-1)) + 4((3)(1) - (1)(-1)))

Simplifying further:

det(A) = (2(-7) - 0(-5) + 4(4))

det(A) = (-14 + 0 + 16)

det(A) = 2

Step 3: Find the matrix of minors. For each element, calculate the determinant of the submatrix obtained by removing the row and column of that element. In our case:

|((-2) (-2)) (5 (-2)) (5 1)|
|((1 -2) (-2 -2)) ((-1 -2) (5 -2)) ((-1 1) (5 -2))|
|((1 1) (-2 1)) ((-1 1) (5 1)) ((-1 1) (5 1))|

Note: Each entry in this matrix is the determinant of a 2x2 matrix.

Calculating these determinants:

Matrix of minors:

|((-2)(-2) - (5)(-2)) (5(-2) - (5)(1)) (5(1) - (1)(5))|
|((1)(-2) - (-2)(-2)) ((-1)(-2) - (5)(-2)) ((-1)(1) - (5)(-2))|
|((1)(1) - (1)(1)) ((-1)(1) - (5)(1)) ((-1)(1) - (5)(1))|

Simplifying:

Matrix of minors:

|(4 -10 -4)|
|(2 -8 -9)|
|(0 -6 -6)|

Step 4: Find the matrix of cofactors. Multiply each element of the matrix of minors by the corresponding sign. The sign pattern is "+ - + - + - ...", starting with a positive sign for the top-left element. In our case:

Matrix of cofactors:

|(+4 -10 +4)|
|(-2 +8 -9)|
|(0 -6 +6)|

Step 5: Transpose the matrix of cofactors to obtain the adjugate matrix. The adjugate matrix is obtained by swapping the rows and columns of the matrix of cofactors. In this case:

Adjugate matrix:

|4 -2 0|
|-10 8 -6|
|4 -9 6|

Step 6: Finally, find the inverse matrix by multiplying the adjugate matrix by the reciprocal of the determinant:

Inverse matrix:

|4 -2 0|
|-10 8 -6|
|4 -9 6| / 2

|2 -1 0|
|-5 4 -3|
|2 -4 3|

Now let's check if the inverse you provided is correct:

|-7/2 2 -2|
|1/2 0 1|
|2 -1 1|

To simplify the problem, you can multiply each element by 2:

|-7 4 -4|
|1 0 2|
|4 -2 2|

Comparing this with the inverse matrix we obtained, we can see that they are not the same. Therefore, the inverse you provided is not correct.

I hope this explanation helps you understand the process of finding the inverse of a matrix.