Adjugate Matrix: Definition, Formation & Example

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  • 0:03 The Adjugate Matrix
  • 0:29 Building the Adjugate Matrix
  • 5:04 Example
  • 6:10 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we'll calculate the adjugate matrix. This will require finding minors, determinants, and transposes. All of these steps will be carefully explained with an example.

The Adjugate Matrix

Let's say that we're renovating the floor in our kitchen. The clay-like terracotta tiles are catching our attention. This earthenware gives a nice warm feeling to a space, and the matrix shape is a bonus! Makes you think of all the different types of matrices that exist. How about the adjugate matrix?

In this lesson, we'll be looking at how to compute the adjugate matrix by transposing the cofactor matrix. Sounds like we'll need some special operations to finish our floor.

Building the Adjugate Matrix

Why would you need an adjugate matrix? This matrix finds applications when inverting a matrix because the matrix inverse is the adjugate matrix divided by the determinant. Building the adjugate matrix isn't complicated, and it takes just two steps: find the cofactor matrix and then transpose. Let's take a closer look at the each of these steps one at a time.

Step One: Finding the Cofactor Matrix

Imagine having a sheet of tile with 16 numbers on it arranged as a 4x4 matrix, like this one:

4x4 tile of numbers.

We start with the first square in the top-left corner. It has the number 6 in it. This square is in row 1 and column 1. Now, imagine deleting row 1 and column 1. We could visualize a line horizontally crossing out row 1 and a line vertically crossing out column 1 like you can see happening in this matrix:

Deleting row 1 and column 1.

What do we have left? It's still a matrix, but it's smaller. In our example, this smaller matrix has the rows: 3 5 7, 3 5 6, and 2 6 3. If we take the determinant of this smaller matrix, it is called the minor of row 1 and column 1. The determinant of a matrix is a number. In our example, the determinant of this smaller matrix is:

3[5(3) - 6(6)] - 5[3(3) - 6(2)] + 7[3(6) - 5(2)]

= 3(-21) - 5(-3) + 7(8)

= -63 + 15 + 56 = 8

We take this number 8 and store it in a new matrix at the location row 1 and column 1. We repeat this process for each of the locations in the original matrix.

Let's do these steps again. What if we wanted the minor for the location: row 4, column 3? Imagine the smaller matrix after deleting row 4 and column 3, as you can see:

Deleting row 4 and column 3.

The remaining matrix has the rows 6 4 8, 1 3 7, and 2 3 6. The determinant is:

6[3(6) - 7(3)] - 4[1(6) - 7(2)] + 8[1(3) - 3(2)] = -10

We store this -10 in the new matrix at the location row 4 and column 3. The new matrix, so far, looks like this matrix:

Two entries in the new matrix.

Before filling in the remaining entries, let's place the minor for row 2, column 4. This number is the determinant of the matrix with rows 6 4 2, 2 3 5, and 4 2 6. This determinant is 64. The updated matrix looks like this:

Three entries in the new matrix.
matrix 3

Looking ahead, there will be a 'sign change step' and a 'transpose step.' It will be easier to see these steps with the partially filled matrix. It's like reviewing a floor plan after some of the new tiles are installed. Now we're ready to move on to our second step in the process of building our adjugate matrix.

Step Two: Sign Changing and Transposing

The cofactor matrix is very close to this new matrix we've been building. All we have to do is multiply each entry by a +1 or by a -1. The plus and minus ones alternate, as you can see:

The alternating signs.

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