# The problem simply is this:

Evaluate: 12 5 -2
-3 0 1
-5 4 2

These figures are in matrix form. I don't understand what it is I am supposed to be evaluating.

## 12 5 -2

-3 0 +1
-5 4 +2

I am going to make a wild guess that it is the determinant they are after. It could be other things, such as inverse, but probably determinant.
There are a lot of ways to find the determinant.

Perhaps the safest is to use a rule that works for any size.
Go element by element down the first column
the determinant is:
first element * determinant of matrix remaining without that row and column
- second element * determinant of remaining matrix without that row and column
= third element * determinant of remaining matrix without that row and column
continue + and - until you reach the bottom
So here we have
12 times the det of
0 1
4 2

+3 times the det of
5 -2
4 +2

+5 times the det of
5 -2
0 +1
Now I will do that first one,
0 1
4 2
using our rule it is 0 (det of remaining matrix which is 2) - 4 (det of remaing matrix which is 1)
= 0*2 - 4*1 = -4
so
we have 12*-4 so far =-48
now we need the second sub matrix determinant to multiply by +3
5 -2
4 +2 Do it yourself and get 18 so for the second prow we get 3*18 = 54
for the third submatrix I got det = 5 and -5 *5 = -25
So , in the end I have
-48 +54 -25 = -19 <--- answer
Now there are easier ways to do a three by three square matrix, but I am playing it safe so you can do other types.

## the determinant is:

first element * determinant of matrix remaining without that row and column
- second element * determinant of remaining matrix without that row and column
+ third element * determinant of remaining matrix without that row and column
continue + and - until you reach the bottom

5 -2
0 +1

## To evaluate a matrix, you need to perform certain operations on its elements. In this case, you are given a 3x3 matrix:

12 5 -2
-3 0 1
-5 4 2

To evaluate this matrix, you can perform different operations depending on what you are looking for. Some common operations include finding the determinant, the transpose, or the inverse of the matrix. Let's go through each of them:

1. Determinant: To find the determinant of a matrix, you can use various methods such as cofactor expansion or row/column operations. The determinant of a 3x3 matrix can be calculated using the following formula:

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

where a11, a12, a13, a21, a22, a23, a31, a32, and a33 represents the elements of the matrix in their respective positions.

By substituting the values from the given matrix into the determinant formula, you can calculate the determinant.

2. Transpose: The transpose of a matrix is obtained by swapping its rows with columns. For example, for a 3x3 matrix A, the transpose matrix A^T will have the elements with the following arrangement:

12 -3 -5
5 0 4
-2 1 2

Each element in the original matrix moves to a new position in the transposed matrix.

3. Inverse: The inverse of a matrix A is denoted as A^(-1) and is such that when you multiply the matrix A with its inverse A^(-1), you get the identity matrix, usually represented by I. The inverse of a 3x3 matrix can be calculated using various methods, such as row operations, adjugate matrix, or using the formula:

A^(-1) = (1 / det(A)) * adj(A)

where det(A) represents the determinant of the matrix A, and adj(A) represents the adjugate matrix of A.

By substituting the values from the given matrix into the inverse formula, you can calculate the inverse matrix.

To evaluate the matrix, you need to specify which operation you want to perform on the matrix, such as finding the determinant, transpose, or inverse.