Definition of A(i ∣ j)

Cofactor expansion

One way of computing the determinant of an n×n matrix A is to use the following formula called the cofactor formula.

Cofactor Formula

Before going to learn the cofactor formula, let us see where it is used. The inverse of a matrix is used in solving a system of equations. The inverse of a matrix is its adjoint divided by its determinant. The adjoint of the matrix is the transpose of the cofactor matrix. The cofactor matrix is the matrix of the same order as the given matrix where the elements of the original matrix are replaced by their corresponding cofactors. These cofactors are calculated using the cofactor formula. Let us learn the cofactor formula along with a few solved examples.

What Is Cofactor Formula?

The cofactor Cij of an element aijaij of a square matrix of order n×n is its minor Mij multiplied by (−1)i+j. Here, the minor Mij of the element aij is the determinant of the matrix obtained by removing the i^th row and j^th column from the original matrix. i.e.,

Cofactor of an element aij is, Cij=(−1)^i+j⋅Mij

Here, Mij = The minor of aij = The determinant of the matrix obtained by removing the i^th row and j^th column.

Examples Using Cofactor Formula

Cofactor Expansions

Objectives

1. Learn to recognize which methods are best suited to compute the determinant of a given matrix.
2. Recipes: the determinant of a 3×3 matrix, compute the determinant using cofactor expansions.
3. Vocabulary words: minor, cofactor.

In this section, we give a recursive formula for the determinant of a matrix, called a cofactor expansion. The formula is recursive in that we will compute the determinant of an n×n matrix assuming we already know how to compute the determinant of an (n−1)×(n−1) matrix.

At the end is a supplementary subsection on Cramer’s rule and a cofactor formula for the inverse of a matrix.

Cofactor Expansions

A recursive formula must have a starting point. For cofactor expansions, the starting point is the case of 1×1 matrices. The definition of determinant directly implies that

det(a)=a.

To describe cofactor expansions, we need to introduce some notation.

Definition

Alternatively, it is not necessary to repeat the first two columns if you allow your diagonals to “wrap around” the sides of a matrix, like in Pac-Man or Asteroids.

Cramer’s Rule and Matrix Inverses

It is clear from the previous example that (4.2.1) is a very inefficient way of computing the inverse of a matrix, compared to augmenting by the identity matrix and row reducing, as in this subsection in Section 3.5. However, it has its uses.

• If a matrix has unknown entries, then it is difficult to compute its inverse using row reduction, for the same reason it is difficult to compute the determinant that way: one cannot be sure whether an entry containing an unknown is a pivot or not.
• This formula is useful for theoretical purposes. Notice that the only denominators in (4.2.1) occur when dividing by the determinant: computing cofactors only involves multiplication and addition, never division. This means, for instance, that if the determinant is very small, then any measurement error in the entries of the matrix is greatly magnified when computing the inverse. In this way, (4.2.1) is useful in error analysis.

The proof of the theorem uses an interesting trick called Cramer’s Rule, which gives a formula for the entries of the solution of an invertible matrix equation.

Cramer’s Rule

Let x=(x1,x2,…,xn) be the solution of Ax=b, where A is an invertible n×n matrix and b is a vector in Rⁿ. Let A_i be the matrix obtained from A by replacing the ith column by b. Then

Solved Examples for You

Question 3: What is meant by cofactor of a matrix?

Answer: A cofactor refers to the number you attain on removing the column and row of a particular element existing in a matrix.

Question 4: What is meant by a minor matrix?

Answer: A minor refers to the square matrix’s determinant whose formation takes place by deleting one column and one row from some larger square matrix.

Question 5: Can we say that the adjoint is the same as the reverse?

Answer: The adjoint of a matrix is also known as the adjugate of a matrix. It refers to the transpose of the cofactor matrix of that particular matrix. For a matrix A, the denotation of adjoint is as adj (A). On the other hand, the inverse of a matrix A refers to a matrix which on multiplication by matrix A, results in an identity matrix.

Question 6: What is meant by rank of a matrix?

Answer: The rank of a matrix refers to the maximum number of linearly independent column vectors that exist in the matrix. Furthermore, it is also the maximum number of linearly independent row vectors that exist in the matrix.

Proof

Let us review what we actually proved in Section 4.1. We showed that if det:{n×n matrices}→R is any function satisfying the four defining properties of the determinant, Definition 4.1.1 in Section 4.1, (or the three alternative defining properties, Remark: Alternative defining properties,), then it also satisfies all of the wonderful properties proved in that section. In particular, since det can be computed using row reduction by Recipe: Computing Determinants by Row Reducing, it is uniquely characterized by the defining properties. What we did not prove was the existence of such a function, since we did not know that two different row reduction procedures would always compute the same answer.

Consider the function dd defined by cofactor expansion along the first row:

If we assume that the determinant exists for (n−1)×(n−1) matrices, then there is no question that the function dd exists, since we gave a formula for it. Moreover, we showed in the proof of Theorem 4.2.1 above that dd satisfies the three alternative defining properties of the determinant, again only assuming that the determinant exists for (n−1)×(n−1) matrices. This proves the existence of the determinant for n×n matrices!

This is an example of a proof by mathematical induction. We start by noticing that det(a)=a satisfies the four defining properties of the determinant of a 1×1 matrix. Then we showed that the determinant of n×n matrices exists, assuming the determinant of (n−1)×(n−1) matrices exists. This implies that all determinants exist, by the following chain of logic:

1×1 exists ⟹ 2×2 exists ⟹ 3×3 exists ⟹ ⋯.

Example 4.2.2

Find the determinant of

which agrees with the formulas in Definition 3.5.2 in Section 3.5 and Example 4.1.6 in Section 4.1.

Alternatively, it is not necessary to repeat the first two columns if you allow your diagonals to “wrap around” the sides of a matrix, like in Pac-Man or Asteroids.

Cofactor expansions are most useful when computing the determinant of a matrix that has a row or column with several zero entries. Indeed, if the (i,j) entry of A is zero, then there is no reason to compute the (i,j) cofactor. In the following example we compute the determinant of a matrix with two zeros in the fourth column by expanding cofactors along the fourth column.

We only have to compute two cofactors. We can find these determinants using any method we wish; for the sake of illustration, we will expand cofactors on one and use the formula for the 3×3 determinant on the other.

Expanding along the first column, we compute

Cofactor expansions are also very useful when computing the determinant of a matrix with unknown entries. Indeed, it is inconvenient to row reduce in this case, because one cannot be sure whether an entry containing an unknown is a pivot or not.

It is often most efficient to use a combination of several techniques when computing the determinant of a matrix. Indeed, when expanding cofactors on a matrix, one can compute the determinants of the cofactors in whatever way is most convenient. Or, one can perform row and column operations to clear some entries of a matrix before expanding cofactors, as in the previous example.

Note 4.2.24.2.2: Summary: Methods for Computing Determinants

We have several ways of computing determinants:

1. Special formulas for 2×2 and 3×3 matrices.
   This is usually the best way to compute the determinant of a small matrix, except for a 3×3 matrix with several zero entries.
2. Cofactor expansion.
   This is usually most efficient when there is a row or column with several zero entries, or if the matrix has unknown entries.
3. Row and column operations.
   This is generally the fastest when presented with a large matrix which does not have a row or column with a lot of zeros in it.
4. Any combination of the above.
   Cofactor expansion is recursive, but one can compute the determinants of the minors using whatever method is most convenient. Or, you can perform row and column operations to clear some entries of a matrix before expanding cofactors.

Remember, all methods for computing the determinant yield the same number.

Cramer’s Rule and Matrix Inverses

Recall from Proposition 3.5.1 in Section 3.5 that one can compute the determinant of a 2×2 matrix using the rule

It is clear from the previous example that (4.2.1) is a very inefficient way of computing the inverse of a matrix, compared to augmenting by the identity matrix and row reducing, as in Subsection Computing the Inverse Matrix in Section 3.5. However, it has its uses.

• If a matrix has unknown entries, then it is difficult to compute its inverse using row reduction, for the same reason it is difficult to compute the determinant that way: one cannot be sure whether an entry containing an unknown is a pivot or not.
• This formula is useful for theoretical purposes. Notice that the only denominators in (4.2.1) occur when dividing by the determinant: computing cofactors only involves multiplication and addition, never division. This means, for instance, that if the determinant is very small, then any measurement error in the entries of the matrix is greatly magnified when computing the inverse. In this way, (4.2.1) is useful in error analysis.

The proof of Theorem 4.2.2 uses an interesting trick called Cramer’s Rule, which gives a formula for the entries of the solution of an invertible matrix equation.

Theorem 4.2.3: Cramer’s Rule

Let x=(x1,x2,…,xn) be the solution of Ax=b, where A is an invertible n×n matrix and b is a vector in Rⁿ. Let A_i be the matrix obtained from AA by replacing the i^th column by bb. Then

and thus

Determinant formulas and cofactors

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