We define $3\times 3$ determinants in terms of $2\times 2$ determinants, $4\times 4$ determinants in terms of $3\times 3$ determinants, and so on. This type of definition—one dimension at a time, with the definition in each dimension depending on its meaning in lower dimensions—is called an inductive definition.

The determinant $\det \mathbf{\text{A}}=\left|a_{ij}\right|$ of a $3\times 3$ matrix $\mathbf{\text{A}}=[a_{ij}]$ is defined as follows:

Note the single minus sign on the right-hand side. The three $2\times 2$ determinants in (5) are multiplied by the elements $a_{11},a_{12},$ and $a_{13}$ along the first row of the matrix A. Each of these elements $a_{1j}$ is multiplied by the determinant of the $2\times 2$ submatrix of A that remains after the row and column containing $a_{1j}$ are deleted.

The definition of higher-order determinants is simplified by the following notation and terminology.

For example, the minor of $a_{12}$ in a $3\times 3$ matrix is

The minor of $a_{32}$ in a $4\times 4$ matrix is

According to Eq. (6), the cofactor $A_{ij}$ is obtained by attaching the sign ${(-1)}^{i+j}$ to the minor $M_{ij}.$ This sign is most easily remembered as the one that appears in the ijth position in checkerboard arrays such as

Note that a plus sign always appears in the upper left corner and that the signs alternate horizontally and vertically. In the $4\times 4$ case, for instance,

and so forth.

With this notation, the definition of $3\times 3$ determinants in (5) can be rewritten as

The last formula is the cofactor expansion of det A along the first row of A. Its natural generalization yields the definition of the determinant of an $n\times n$ matrix, under the inductive assumption that $(n-1)\times (n-1)$ determinants have already been defined.

In numerical computations, it frequently is more convenient to work first with minors rather than with cofactors and then attach signs in accord with the checkerboard pattern illustrated previously. Note that determinants have been defined only for square matrices.

Note that, if we could expand along the second row in Example 4, there would be only a single $3\times 3$ determinant to evaluate. It is in fact true that a determinant can be evaluated by expansion along any row or column. The proof of the following theorem is included in Appendix B.

The formulas in (9) and (10) provide 2n different cofactor expansions of an $n\times n$ determinant. For $n=3,$ for instance, we have

In a specific example, we naturally attempt to choose the expansion that requires the least computational labor.

In addition to providing ways of evaluating determinants, the theorem on cofactor expansions is a valuable tool for investigating the general properties of determinants. For instance, it follows immediately from the formulas in (9) and (10) that, if the square matrix A has either an all-zero row or an all-zero column, then $\det \mathbf{\text{A}}=0.$ For example, by expanding along the second row we see immediately that

We now list seven properties of determinants that simplify their computation. Each of these properties is readily verified directly in the case of $2\times 2$ determinants. Just as our definition of $n\times n$ determinants was inductive, the following discussion of Properties 1–7 of determinants is inductive. That is, we suppose that $n\ge 3$ and that these properties have already been verified for $(n-1)\times (n-1)$ determinants.

Property 1: If the $n\times n$ matrix B is obtained from A by multiplying a single row (or a column) of A by the constant k, then $\det \mathbf{\text{B}}=k\det \mathbf{\text{A}}$.

For instance, if the ith row of A is multiplied by k, then the elements off the ith row of A are unchanged. Hence for each $j=1,2,\dots ,n,$ the ijth cofactors of A and B are equal: $A_{ij}=B_{ij}.$ Therefore, expansion of B along the ith row gives

and thus $\det \mathbf{\text{B}}=k\det \mathbf{\text{A}}$.

Property 1 implies simply that a constant can be factored out of a single row or column of a determinant. Thus we see that

by factoring 3 out of the second column.

Property 2: If the $n\times n$ matrix B is obtained from A by interchanging two rows (or two columns), then $\det \mathbf{\text{B}}=-\det \mathbf{\text{A}}$.

To see why this is so, suppose (for instance) that the first row is not one of the two that are interchanged (recall that $n\ge 3$). Then for each $j=1,2,\dots ,n,$ the cofactor $B_{1j}$ is obtained by interchanging two rows of the cofactor $A_{1j}.$ Therefore, $B_{1j}=-A_{1j}$ by Property 2 for $(n-1)\times (n-1)$ determinants. Because $b_{1j}=a_{1j}$ for each j, it follows by expanding along the first row that

and thus $\det \mathbf{\text{B}}=-\det \mathbf{\text{A}}$.

Property 3: If two rows (or two columns) of the $n\times n$ matrix A are identical, then $\det \mathbf{\text{A}}=0$.

To see why, let B denote the matrix obtained by interchanging the two identical rows of A. Then $\mathbf{\text{B}}=\mathbf{\text{A}},$ so $\det \mathbf{\text{B}}=\det \mathbf{\text{A}}.$ But Property 2 implies that $\det \mathbf{\text{B}}=-\det \mathbf{\text{A}}.$ Thus $\det \mathbf{\text{A}}=-\det \mathbf{\text{A}},$ and it follows immediately that $\det \mathbf{\text{A}}=0$.

Property 4: Suppose that the $n\times n$ matrices ${\mathbf{\text{A}}}_1,{\mathbf{\text{A}}}_2,$ and B are identical except for their ith rows—that is, the other $n-1$ rows of the three matrices are identical—and that the ith row of B is the sum of the ith rows of ${\mathbf{\text{A}}}_1$ and ${\mathbf{\text{A}}}_2.$ Then

This result also holds if columns are involved instead of rows.

Property 4 is readily established by expanding B along its ith row. In Problem 45 we ask you to supply the details for a typical case. The main importance (at this point) of Property 4 is that it implies the following property relating determinants and elementary row operations.

Property 5: If the $n\times n$ matrix B is obtained by adding a constant multiple of one row (or column) of A to another row (or column) of A, then $\det \mathbf{\text{B}}=\det \mathbf{\text{A}}$.

Thus the value of a determinant is not changed either by the type of elementary row operation described or by the corresponding type of elementary column operation. The following computation with $3\times 3$ matrices illustrates the general proof of Property 5. Let

So B is the result of adding k times the first column of A to its third column. Then

Here we first applied Property 4 and then factored k out of the second summand with the aid of Property 1. Now the first determinant on the right-hand side in (11) is simply $\det \mathbf{\text{A}},$ whereas the second determinant is zero (by Property 3—its first and third columns are identical). We therefore have shown that $\det \mathbf{\text{B}}=\det \mathbf{\text{A}},$ as desired.

Although we use only elementary row operations in reducing a matrix to echelon form, Properties 1, 2, and 5 imply that we may use both elementary row operations and the analogous elementary column operations in simplifying the evaluation of determinants.

An upper triangular matrix is a square matrix having only zeros below its main diagonal. A lower triangular matrix is a square matrix having only zeros above its main diagonal. A triangular matrix is one that is either upper triangular or lower triangular, and thus looks like

The next property tells us that determinants of triangular matrices are especially easy to evaluate.

Property 6: The determinant of a triangular matrix is equal to the product of its diagonal elements.

The reason is that the determinant of any triangular matrix can be evaluated as in the following computation:

At each step we expand along the first column and pick up another diagonal element as a factor of the determinant.

The transpose ${\mathbf{\text{A}}}^T$ of a $2\times 2$ matrix A is obtained by interchanging its off-diagonal elements:

More generally, the transpose of the $m\times n$ matrix $\mathbf{\text{A}}=[a_{ij}]$ is the $n\times m$ matrix ${\mathbf{\text{A}}}^T$ defined by

Note the reversal of subscripts; this means that the element of ${\mathbf{\text{A}}}^T$ in its jth row and ith column is the element of A in the ith row and the jth column of A. Hence the rows of the transpose ${\mathbf{\text{A}}}^T$ are (in order) the columns of A, and the columns of ${\mathbf{\text{A}}}^T$ are the rows of A. Thus we obtain ${\mathbf{\text{A}}}^T$ by changing the rows of A into columns. For instance,

and

If the matrix A is square, then we get ${\mathbf{\text{A}}}^T$ by interchanging elements of A that are located symmetrically with respect to its main diagonal. Thus ${\mathbf{\text{A}}}^T$ is the mirror reflection of A through its main diagonal.

In Problems 47 and 48 we ask you to verify the following properties of the transpose operation (under the assumption that A and B are matrices of appropriate sizes and c is a number):

1. ${({\mathbf{\text{A}}}^T)}^T=\mathbf{\text{A}}$;

2. ${(\mathbf{\text{A}}+\mathbf{\text{B}})}^T={\mathbf{\text{A}}}^T+{\mathbf{\text{B}}}^T$;

3. ${(c\mathbf{\text{A}})}^T=c{\mathbf{\text{A}}}^T$;

4. ${(\mathbf{\text{AB}})}^T={\mathbf{\text{B}}}^T{\mathbf{\text{A}}}^T$.

Property 7: If A is a square matrix, then $\det ({\mathbf{\text{A}}}^T)=\det \mathbf{\text{A}}$.

This property of determinants can be verified by writing the cofactor expansion of det A along its first row and the cofactor expansion of $\det ({\mathbf{\text{A}}}^T)$ along its first column. When this is done, we see that the two expansions are identical, because the rows of A are the columns of ${\mathbf{\text{A}}}^T$ (as in Problem 49).

We began this section with the remark that a $2\times 2$ matrix A is invertible if and only if its determinant is nonzero: $\left|\mathbf{\text{A}}\right|\ne 0.$ This result holds more generally for $n\times n$ matrices.

This theorem (along with the others in this section) is proved in Appendix B. So now we can add the statement that $\det \mathbf{\text{A}}\ne 0$ to the list of equivalent properties of nonsingular matrices stated in Theorem 7 of Section 3.5. Indeed, some texts define the square matrix A to be nonsingular if and only if $\det \mathbf{\text{A}}\ne 0$.

The connection between nonzero determinants and matrix invertibility is closely related to the fact that the determinant function “respects” matrix multiplication in the sense of the following theorem.

The fact that $\left|\mathbf{\text{AB}}\right|=\left|\mathbf{\text{A}}\right|\left|\mathbf{\text{B}}\right|$ seems so natural that we might fail to note that it is also quite remarkable. Contrast the simplicity of the equation $\left|\mathbf{\text{AB}}\right|=\left|\mathbf{\text{A}}\right|\left|\mathbf{\text{B}}\right|$ with the complexity of the seemingly unrelated definitions of determinants and matrix products. For another contrast, we can mention that $|\mathbf{\text{A}}+\mathbf{\text{B}}|$ is generally not equal to $\left|\mathbf{\text{A}}\right|+\left|\mathbf{\text{B}}\right|.$ (Pick a pair of $2\times 2$ matrices and calculate their determinants and that of their sum.)

As a first application of Theorem 3, we can calculate the determinant of the inverse of an invertible matrix A:

so

and therefore

Now $\left|\mathbf{\text{A}}\right|\ne 0$ because A is invertible. Thus $\det ({\mathbf{\text{A}}}^{-1})$ is the reciprocal of det A.

Suppose that we need to solve the $n\times n$ linear system

where

We assume that the coefficient matrix A is invertible, so we know in advance that a unique solution x of (16) exists. The question is how to write x explicitly in terms of the coefficients $a_{ij}$ and the constants $b_i.$ In the following discussion, we think of x as a fixed (though as yet unknown) vector.

If we denote by ${\mathbf{\text{a}}}_1,{\mathbf{\text{a}}}_2,\dots ,{\mathbf{\text{a}}}_n$ the column vectors of the $n\times n$ matrix A, then

The desired formula for the ith unknown $x_i$ involves the determinant of the matrix $[\begin{array}{lllll}{\mathbf{a}}_{\mathbf{1}}& \cdots & \mathbf{b}& \cdots & {\mathbf{a}}_n\end{array}]$ that is obtained upon replacing the ith column ${\mathbf{\text{a}}}_i$ of A with the constant vector b.

For instance, the unique solution $(x_1,x_2,x_3)$ of the $3\times 3$ system

with $\left|\mathbf{\text{A}}\right|=\left|a_{ij}\right|\ne 0$ is given by the formulas

The inverse ${\mathbf{\text{A}}}^{-1}$ of the invertible $n\times n$ matrix A can be found by solving the matrix equation

If we write the coefficient matrix $\mathbf{\text{A}}=\left[\begin{array}{llll}{\mathbf{\text{a}}}_1& {\mathbf{\text{a}}}_2& \cdots & {\mathbf{\text{a}}}_n\end{array}\right],$ the unknown matrix $\mathbf{\text{X}}=\left[\begin{array}{llll}{\mathbf{\text{x}}}_1& {\mathbf{\text{x}}}_2& \cdots & {\mathbf{\text{x}}}_n\end{array}\right],$ and the identity matrix $\mathbf{\text{I}}=\left[\begin{array}{llll}{\mathbf{\text{e}}}_1& {\mathbf{\text{e}}}_2& \cdots & {\mathbf{\text{e}}}_n\end{array}\right]$ in terms of their columns, then (20) says that

for $j=1,2,\dots ,n.$ Each of these n equations can then be solved explicitly using Cramer’s rule. In particular, Eq. (17) says that the ith element of the column matrix ${\mathbf{\text{x}}}_j$ is given by

for i, $j=1,2,\dots ,n.$ When these results are assembled (Appendix B), we get an explicit formula for the inverse matrix.

We see in (23) the transpose of the cofactor matrix $[A_{ij}]$ of the $n\times n$ matrix A. This transposed cofactor matrix is called the adjoint matrix of A and is denoted by

The amount of labor required to complete a numerical calculation is measured by the number of arithmetical operations it involves. So let us count just the number of multiplications required to evaluate an $n\times n$ determinant using cofactor expansions. If $n=5,$ for instance, then the cofactor expansion along a row or column requires computations of five $4\times 4$ determinants. A cofactor expansion of each of these five $4\times 4$ determinants involves four $3\times 3$ determinants. Each of these four $3\times 3$ determinants leads to three $2\times 2$ determinants, and finally each of these $2\times 2$ determinants requires two multiplications for its evaluation. Hence the total number of multiplications required to evaluate the original $5\times 5$ determinant is

In general, the number of multiplications required to evaluate an $n\times n$ determinant completely by cofactor expansions is

Thus, if we ignore the smaller number of additions involved, then our operations count for an $n\times n$ determinant is n!. For instance, the evaluation of a $25\times 25$ determinant by cofactor expansion would require about $25!\approx 1.55\times {10}^{25}$ operations. If we used a supercomputer capable of a billion operations per second this task would require about $(1.55\times {10}^{16})/(365.25\times 24\times 3600)\approx \text{492 million years!}$ This same supercomputer would require about $9.64\times {10}^{47}$ years—incomparably longer than the estimated age of the universe (perhaps 20 billion years)—to evaluate a $50\times 50$ determinant by cofactor expansion. Contemporary scientific and engineering applications routinely involve matrices of size $1000\times 1000$ (or larger) whose cofactor expansions would require incomprehensibly long times with any conceivable computer.

However, a typical 2000-era desktop computer, using Matlab and performing “only” about 100 million operations per second, calculates the determinant or inverse of a randomly generated $100\times 100$ matrix almost instantaneously. How is this possible? Obviously not by using cofactor expansions!

The answer is that determinants are much more efficiently calculated by Gaussian elimination—that is, by reduction to triangular form, so that only the product of the remaining diagonal elements need be calculated. Similarly, the inverse of the invertible matrix A is much more efficiently calculated by reduction of the augmented matrix $\left[\begin{array}{ll}\mathbf{\text{A}}& \mathbf{\text{I}}\end{array}\right]$ to reduced echelon form (as in Section 3.5) than by use of Cramer’s rule (as in Theorem 5 here). A careful count reveals that the number of arithmetic operations required to reduce an $n\times n$ matrix to echelon form is of the order of $n^3$ rather than $n!.$ As indicated in the table of Fig. 3.6.1, $n^3$ is dramatically smaller than $n!$ if n is fairly large. Consequently, cofactor expansions of determinants and Cramer’s rule for the solution of systems are primarily of theoretical importance, and are seldom used in numerical problems where n is greater than 3 or 4.