Cofactor Expansion

Theorem: If square matrix A is invertible;

In another lesson on Determinants and Inverses, we learned that the inverse
of a 2 × 2 matrix is found using the algorithm which states:

If det (A) is not = 0, then A ^{– 1} exists and is equal to

Let's compare the two formulas.
They both have (1 over det A) times a matrix called the Adjoint of A.

We got the 2 × 2 inverse matrix by:

1) switching the main diagonal entries of A
2) negating the minor diagonal entries of A.

If the two formulae are the same, this matrix must be the Adjoint matrix

Definition: the adjoint of matrix A is the transpose of the matrix of cofactors.

Since we know how to transpose a matrix, all we need to know now is:
"what the heck are cofactors and how do we find them?"
Definition: the cofactor C_{i j} of entry a_{i j} is (-1)^{i+ j}(M_{i j}).

Definition: the minor M_{i j} of entry a_{i j} is the determinant of the matrix
that remains after we eliminate the ith row and jth column of matrix A.

So, to generate the matrix of cofactors,

1) we choose an entry a_{i j}
2) find i + j to see whether we get a + or – from ( – 1)^{i+ j}
3) find the minor M_{i j}.

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Example 1

Find the matrix of cofactors if

The matrix of cofactors then is: , and its transpose is

Since det A = 5, the inverse matrix A ^{– 1} = is

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Now let's see how this applies to a 2 × 2 matrix.
When we eliminate the ith row and jth column of a 2 × 2, we
eliminate 3 of the 4 entries in the matrix, so we're left with a single entry.
That entry is the cofactor we're seeking. Let's do one with this method.

Example 2

Using cofactors, find the inverse of .

Determinant = 2.

C11 = 2,

C12 = – 3

C21 = 0

C22 = 1

So, the inverse of which is what the algorithm gives us.

Note: To save time, I generally go right to the Adjoint by entering the cofactors
in transpose positions. I enter the cofactors of the 1st row in the 1st column, etc.
to avoid the step of writing the matrix of cofactors and then transposing it.
Try it, you might like it. It speeds up the process.

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Practice

1) Use the determinant and the adjoint of A to find A^-1 if

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2) Using the answer from 1, solve this system.

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Solutions

1)det A = –1

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2) Using X = A ^{– 1}B, we get

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