can someone explain to me this one as well step by step...thanks

Problem #2

Solve the matrix equation AX=B for X by finding A to power.

A=
2_-1_3
-3_4_-5
-4_0_6

b=
9
6
-9

Solution is:

5.43
8.22
2.12

To solve the matrix equation AX = B, where A is a matrix, X is the unknown matrix, and B is a known matrix, we can use the inverse of matrix A.

Here are the steps to solve the equation:

Step 1: Calculate the inverse of matrix A:
To find the inverse of matrix A, we need to calculate its determinant (denoted as |A|) and the adjugate of A (denoted as adj(A)).

- Calculate the determinant of A:
To calculate the determinant of a 3x3 matrix, we can use the formula |A| = a(ei - fh) - b(di - fg) + c(dh - eg), where a, b, c, d, e, f, g, h, and i are the elements of matrix A.

In this case, A =
2 -1 3
-3 4 -5
-4 0 6

So, |A| = 2(4*6 - 0*(-5)) - (-1)(-3*6 - 0*(-5)) + 3(-3*0 - 4*(-5))
= 2(24) - (-3)(-18) + 3(-20)
= 48 + 54 - 60
= 42

- Calculate the adjugate of matrix A:
To find the adjugate of a 3x3 matrix, we need to calculate the transpose of the matrix of cofactors.

The matrix of cofactors can be calculated by multiplying each element of matrix A by its corresponding cofactor sign (+ or -).

Cofactor sign: +,-,+, -,+, -,+, -,+
For example, the cofactor sign of element 2 in row 1 column 1 is +, the cofactor sign of element -3 in row 1 column 2 is -, and so on.

Once you have the matrix of cofactors, take its transpose to obtain the adjugate of matrix A.

In this case, the matrix of cofactors of A is:
(4 6)
(5 6)
(-4 3)

Taking the transpose, we get the adjugate of A as:
4 5 -4
6 6 3

Step 2: Calculate X using the formula X = A^(-1)B.
To calculate X, multiply the inverse of matrix A by matrix B.

Using the inverse of A calculated in Step 1 and the given B = (9, 6, -9), we can calculate X.

X = A^(-1)B
= adj(A)/|A| * B

(5.43)
(8.22)
(2.12)

So, the solution to the matrix equation AX = B is X =
5.43
8.22
2.12