Can someone explain to me the following step by step , i 've looked in websites , plus the book and i want to understand how this problem has this answer.

Problem #1

Find the inverse if it exists, for the matrix.

-2_1_3
3_-1_0
-4_2_0

Answer is:
-4_6_5
-8_12_13
2__0__-1

sorry the problem posted up twice

To find the multiplicative inverse, you can use this form:

-2..1..3..|..1..0..0
.3.-1..0..|..0..1..0
-4..2..0..|..0..0..1

The objective is to get the left side to look like the right side, and the multiplicative inverse will be on the right side when finished. Whatever operations you do to the left side, you need to do to the right side as well.

Here's a few basic row operations:
1. Any 2 rows can be interchanged.
2. Any row can be multiplied by a number other than zero.
3. Any row can be replaced by adding a multiple of another row plus that row.

If the answer is as you stated, you should end up with this:

1..0..0..|.-4..6..5
0..1..0..|.-8.12.13
0..0..1..|..2..0.-1

I haven't worked this out, so I hope this will help get you started.

To find the inverse of a matrix, you can use the method of row operations known as Gauss-Jordan elimination. Here are the steps to find the inverse of the given matrix:

Step 1: Write down the given matrix augmented with the identity matrix:

-2..1..3..|..1..0..0
.3.-1..0..|..0..1..0
-4..2..0..|..0..0..1

Step 2: Perform row operations to get the left side to look like the right side:

-2..1..3..|..1..0..0
.3.-1..0..|..0..1..0
-4..2..0..|..0..0..1

Multiply the first row by 1/(-2) to make the leading entry (-2) in the first row equal to 1:

1...-1/2..-3/2..|..-1/2..0..0
.3....-1....0....|....0...1....0
-4.....2....0....|....0...0....1

Add 2 times the first row to the second row to eliminate the entry in the second row and first column:

1...-1/2..-3/2..|..-1/2..0...0
..0..1/2....3/2..|..3/2...1...0
-4.....2....0....|....0...0....1

Add 4 times the first row to the third row to eliminate the entry in the third row and first column:

1...-1/2..-3/2..|..-1/2..0...0
..0..1/2....3/2..|..3/2...1....0
..0.....0....-6..|...-2...2....1

Step 3: Multiply the second row by 2 to make the leading entry (1/2) in the second row equal to 1:

1...-1/2..-3/2..|..-1/2..0...0
....1.....3......|...3......2.....0
....0.....0....-6..|...-2...2....1

Add 3/2 times the second row to the first row to eliminate the entry in the first row and second column:

1.....0......0......|-4.....6.....5
....1.....3......|...3......2.....0
....0.....0....-6..|...-2...2....1

Add 3 times the second row to the third row to eliminate the entry in the third row and second column:

1.....0......0......|-4.....6.....5
....1.....3......|...3......2.....0
....0.....0......0....|-5..-4....1

Step 4: Multiply the third row by (-1/5) to make the leading entry (-5) in the third row equal to 1:

1.....0......0......|-4.....6.....5
....1.....3......|...3......2.....0
....0.....0......1....|1....4....-1/5

Add 3 times the third row to the second row:

1.....0......0......|-4.....6.....5
....1.....3......0....|6....14....-1/5
....0.....0......1....|1....4....-1/5

Step 5: Subtract 3 times the third row from the first row:

1.....0.....0......|..-4.....6.....5
....1.....3......0....|6....14....-1/5
....0.....0......1....|1......4....-1/5

Subtract 3 times the third row from the second row:

1.....0.....0......|..-4.....6.....5
....1.....3......0....|3.....2.....-4/5
....0.....0......1....|......1......-1/5

The augmented matrix on the left side has been transformed into the identity matrix, and the right side contains the inverse of the original matrix:

-4...6...5
...-8..12..13
....2...0..-1

Therefore, the inverse of the given matrix is:

-4..6..5
-8..12..13
..2...0..-1