Mathematics 270 Linear Algebra I

Study Guide :: Unit 3

Determinant of a Matrix

In this unit, we introduce the concept and properties of the determinant of a square matrix $A$, denoted by $\det \left(A\right)$, and study its relationship to inverse matrices and solutions of systems of linear equations. In particular, we generalize the results of Theorem 2 of Section 2.1 to $n\times n$ matrices and also extend the Equivalent Statements Theorem 5 presented in Section 2.3 with one more statement.

Objectives

After completing Unit 3, you should be able to:

1. find the minor and the cofactor of any entry of a square matrix;
2. calculate the determinant of a square matrix using cofactor expansion;
3. calculate the determinant of triangular matrices (upper and lower) and of diagonal matrices by inspection;
4. understand the effect of elementary row operations on the determinant of a matrix;
5. calculate the determinant of a square matrix using row reduction;
6. be able to use a combination of row reduction and cofactor expansion to calculate determinants;
7. understand the Equivalent Statements Theorem, which includes using the determinant of a matrix for testing its invertibility;
8. know the definition of the adjoint of a square matrix;
9. compute the inverse of a matrix using the adjoint of the matrix; and
10. solve a system of linear equations using Cramer’s Rule.

3.1  Defining the Determinant with Minors and Cofactors

This section introduces the concepts of minors and cofactors, which are then used to provide the definition of the determinant of an $n\times n$ matrix. Before going through the corresponding textbook reading let’s intuit minors and cofactors and how they relate to the definition of a determinant.

In Section 2.1, we defined the determinant of a $2\times 2$ matrix

$A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)$ (3.1)

as

$\det \left(A\right)=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|=a_{11}{a}_{22}-a_{12}{a}_{21}$. (3.2)

Notice that if we also define the determinant of a $1\times 1$ matrix $a$ (i.e., a real number $a$) as

$\det \left(a\right)=a$

then we can express the determinant of a $2\times 2$ matrix as

$\det \left(A\right)=\det \left(a_{11}\right)\det \left(a_{22}\right)-\det \left(a_{12}\right)\det \left(a_{21}\right)$,

or as

$\det \left(A\right)=a_{11}\det \left(a_{22}\right)-a_{12}\det \left(a_{21}\right)$ (3.3)

We will take this last expression for the determinant of a $2\times 2$ matrix $A$ as the motivation for defining the determinant of the following $3\times 3$ matrix. Specifically, we could think of expression 3.3 for the determinant of the $2\times 2$ matrix in Equation 3.1 as having been obtained with the following procedure:

1. We cross over the first row and the first column of the matrix (3.1):

   $A=\left(\begin{array}{cc}\sout{a_{11}}\hfill & \sout{a_{12}}\\ \overline{)a_{21}}\hfill & a_{22}\end{array}\right)$ (3.4)

2. To obtain the first term of the determinant (3.3), we multiply the entry where the crossing originated, i.e., $a_{11}$ (since we crossed the first row and first column), multiplied by the determinant of what remains after the crossing, i.e., $\det \left(a_{22}\right)$. This gives us the first term $a_{11}\det \left(a_{22}\right)$ in Equation (3.3) or, in other words, the boxed term in the following expression:

   $\det \left(A\right)=\boxed{a_{11}\det \left(a_{22}\right)}-a_{12}\det \left(a_{21}\right)$.

3. We cross over the first row and the second column of the matrix (3.1):

   $A=\left(\begin{array}{cc}\sout{a_{11}}& \sout{a_{12}}\\ a_{21}& \overline{)a_{22}}\end{array}\right)$ (3.5)

4. We obtain the second term of the determinant (3.3) by multiplying the entry where the crossing originated, i.e., $a_{12}$ (since we crossed the first row and second column), multiplied by the determinant of what remains after the crossing, i.e., $\det \left(a_{21}\right)$. This gives us the second term $a_{12}\det \left(a_{21}\right)$ in Equation (3.3) or, in other words, the boxed term in the following expression:

   $\det \left(A\right)=a_{11}\det \left(a_{22}\right)-\boxed{a_{12}\det \left(a_{21}\right)}$.

5. We obtain the signs in front of each of these terms by adding the number of the row to the number of the column that we eliminated to obtain that term. The sign will be positive if this addition is even and will be negative if the addition is odd. Thus, the sign of the first term (obtained from eliminating the first row and first column) is positive, since $1+1=2$ is even, and the sign of the second term (obtained from eliminating the first row and second column) is negative, since $1+2=3$ is odd. Therefore, we have the following signs (illustrated within boxes) for Equation (3.3):

   $\det \left(A\right)=\boxed{+}{a}_{11}\det \left(a_{22}\right)\boxed{-}{a}_{12}\det \left(a_{21}\right)$.

Note that these steps provide a procedure to find the determinant of a $2\times 2$ matrix. If we now reproduce these steps for the following $3\times 3$ matrix

$A=\left(\begin{array}{lll}{a}_{11}\hfill & a_{12}\hfill & a_{13}\hfill \\ a_{21}\hfill & a_{22}\hfill & a_{23}\hfill \\ a_{31}\hfill & a_{32}\hfill & a_{33}\hfill \end{array}\right)$, (3.6)

we will obtain the following procedure to find an expression for its determinant:

1. We cross over the first row and the first column of the matrix (3.6):

   $A=\left(\begin{array}{ccc}\sout{a_{11}}& \sout{a_{12}}& \sout{a_{13}}\\ \overline{)a_{21}}& a_{22}& a_{23}\\ \overline{)a_{31}}& a_{32}& a_{33}\end{array}\right)$ (3.7)

2. We obtain the first term for the determinant by multiplying the entry where the crossing originated, i.e., $a_{11}$, multiplied by the determinant of the submatrix that remains after the crossing (which we call minor of entry $a_{11}$ and denote as $M_{11}$), i.e.,

   $a_{11}\det \left(\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right)=a_{11}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|=a_{11}{M}_{11}$. (3.8)

   The sign of this term is positive, since we are crossing the first row and the first column (and $1+1=2$, which is even). Thus, the determinant of the matrix $A$ in Equation 3.6 so far looks like

   $\det \left(A\right)=a_{11}{M}_{11}+\dots$

3. We cross over the first row and the second column of the matrix (3.6):

   $A=\left(\begin{array}{ccc}\sout{a_{11}}& \sout{a_{12}}& \sout{a_{13}}\\ a_{21}& \overline{)a_{22}}& a_{23}\\ a_{31}& \overline{)a_{32}}& a_{33}\end{array}\right)$ (3.9)

4. We obtain the second term for the determinant by multiplying the entry where the crossing originated, i.e., $a_{12}$, multiplied by the determinant of the submatrix that remains after the crossing (which we call minor of entry $a_{12}$ and denote as $M_{12}$), i.e.,

   $a_{12}\det \left(\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right)=a_{12}\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|=a_{12}{M}_{12}$. (3.10)

   The sign of this term is negative, since we are crossing the first row and the second column (and $1+2=3$, which is odd). Thus, the determinant of 3.6 so far looks like

   $\det \left(A\right)=a_{11}{M}_{11}-a_{12}{M}_{12}+\dots$

5. We cross over the first row and the third column of the matrix (3.6):

   $A=\left(\begin{array}{ccc}\sout{a_{11}}& \sout{a_{12}}& \sout{a_{13}}\\ a_{21}& a_{22}& \overline{)a_{23}}\\ a_{31}& a_{32}& \overline{)a_{33}}\end{array}\right)$ (3.11)

6. We obtain the third term for the determinant by multiplying the entry where the crossing originated, i.e., $a_{13}$, multiplied by the determinant of the submatrix that remains after the crossing (which we call minor of entry $a_{13}$ and denote as $M_{13}$), i.e.,

   $a_{12}\det \left(\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right)=a_{13}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|=a_{13}{M}_{13}$. (3.12)

   The sign of this term is positive, since we are crossing the first row and the third column (and $1+3=4$, which is even). Thus, the determinant of (3.6) will have the following expression:

   $\det \left(A\right)=a_{11}{M}_{11}-a_{12}{M}_{12}+a_{13}{M}_{13}$.

Now, if we define the cofactors as simply the minors with their associated signs, we can rewrite this last expression as

$\det \left(A\right)=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$,

where $C_{11}=+M_{11}$ is the cofactor of entry $a_{11}$, $C_{12}=-M_{12}$ is the cofactor of entry $a_{12}$, and $C_{13}=+M_{13}$ is the cofactor of entry $a_{13}$.

In summary, an expression for the determinant of the $3\times 3$ matrix (3.6) is given by

$\det (A)=a_{11}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|-a_{12}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|+a_{13}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

$=a_{11}{M}_{11}-a_{12}{M}_{12}+a_{13}{M}_{13}$ (3.13)

$=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$.

This expression for the determinant of a $3\times 3$ matrix is called the cofactor expansion of the matrix (3.6) along the first row, and it is only one way to express the determinant of the matrix. It will become apparent after the corresponding reading for this section that this procedure can be applied to any of the rows or any of the columns in order to calculate the determinant of a matrix.

Reading Assignment

Read, and study, pages 105-110 of the textbook (to “Exercise Set 2.1”).

This section of the textbook introduces formally the concepts of minors and cofactors and uses them to define the determinant of an $n\times n$ matrix. The definition, given in page 108, is applied to various examples for the purpose of calculating the determinant, and it is also used to prove that the determinant of a triangular matrix is the product of the entries in the main diagonal.

Exercises

Work on the following textbook exercises from “Exercise Set 2.1” (pp. 111-112):

• 1, 3, 7, 9, 15, 19, 21, 25, 27, 29, 31, 33; and
• “True-False Exercises” (a, b, c, d, e, f, g).

3.2  Determinants by Row Reduction; Properties of the Determinant

The objective of this section is to simplify the task of calculating the determinant of a matrix by reducing it to row echelon form, which has a determinant that is easier to calculate than the original matrix.

Reading Assignment

Read, and study, pages 113-116 of the textbook (to “Exercise Set 2.2”).

After this reading, it will become clear how each of the elementary row operations performed in the row reduction process of a matrix affects its determinant. In particular, we will see that

1. multiplying a row of a matrix by a constant $k$ will affect its determinant by a factor $k$ ;
2. interchanging two rows will change the sign of the determinant; and
3. adding a multiple of a row/column of a matrix to another row/column does not affect its determinant.

Having this knowledge allows us to simplify the task of calculating the determinant of a given matrix by simply keeping track of this effect while reducing the matrix to its echelon form.

Pay close attention to the theorems stated in this section.

Exercises

Work on the following textbook exercises from “Exercise Set 2.2” (pp. 117-118):

• 3, 5, 7, 9, 11, 17, 21, 27, 29; and
• “True-False Exercises” (a, b, c, e).

3.3  More Properties of the Determinant; Adding to the Equivalent Statements Theorem for an Invertible Matrix

This section presents the properties concerning the determinant of a matrix multiplied by a scalar, $\det \left(kA\right)$ ; the determinant of a summation of matrices, $\det \left(A+B\right)$; the determinant of a multiplication of matrices, $\det \left(AB\right)$; and the determinant of the inverse of a matrix, $\det \left(A^{-1}\right)$. In this section, we also add another statement to the Equivalent Statements Theorem for an Invertible Matrix (Theorem 5 in Section 2.3). In particular, we extend the Equivalent Statements Theorem to the following:

Theorem 6.  Equivalent Statements

If $A$ is an $n\times n$ matrix, then the following statements are equivalent (they are all true or all false):

1. $A$ is invertible.
2. $A\overrightarrow{x}=\overrightarrow{0}$ has only the trivial solution.
3. The reduced row echelon form of $A$ is $I_n$.
4. $A$ can be expressed as a product of elementary matrices.
5. $A\overrightarrow{x}=\overrightarrow{b}$ is consistent for every $n\times 1$ matrix $\overrightarrow{b}$.
6. $A\overrightarrow{x}=\overrightarrow{b}$ has exactly one solution for every vector $n\times 1$ matrix $\overrightarrow{b}$.
7. $\det \left(A\right)\ne 0$

Reading Assignment

Read, and study, pages 118-122 of the textbook (to “Adjoint of a Matrix”) and page 126 (Equivalent Statements Theorem).

Exercises

Work on the following textbook exercises from “Exercise Set 2.3” (pp. 127-128):

• 3, 5, 7, 9, 13, 15, 17, 33, 35, 37; and
• “True-False Exercises” (a, b, c, d, g, h, i).

3.4  Cramer’s Rule

This section uses the properties of the determinants studied in the previous sections to derive a formula for the inverse of a matrix. In particular, the formula given by Equation (2.3) in Theorem 2 for the inverse of a $2\times 2$ matrix will be generalized for a general $n\times n$ matrix. The result is then used to find an explicit formula for the solution of a system of linear equations, called Cramer’s Rule, in terms of the determinant of the coefficient matrix.

Reading Assignment

Read, and study, pages 122-126 of the textbook (from “Adjoint of a Matrix” to “Exercise Set 2.3”).

The reading starts by introducing the definition of the adjoint of a matrix $A$, denoted by adj( $A$), which is the transpose of the matrix formed by the cofactors of $A$ ; in other words, if we denote the cofactor matrix as

$C=\left(\begin{array}{cccc}{C}_{11}& C_{12}& \cdots & C_{1n}\\ C_{21}& C_{22}& \cdots & C_{2n}\\ ⋮& ⋮& & ⋮\\ C_{n1}& C_{n2}& \cdots & C_{nn}\end{array}\right)$, (3.14)

where $C_{ij}$ is the cofactor of entry $a_{ij}$, then the adjoint of $A$, is defined as

$\text{adj(}A\text{)}=C^T={\left(\begin{array}{cccc}{C}_{11}& C_{12}& \cdots & C_{1n}\\ C_{21}& C_{22}& \cdots & C_{2n}\\ ⋮& ⋮& & ⋮\\ C_{n1}& C_{n2}& \cdots & C_{nn}\end{array}\right)}^T=\left(\begin{array}{cccc}{C}_{11}& C_{21}& \cdots & C_{n1}\\ C_{12}& C_{22}& \cdots & C_{n2}\\ ⋮& ⋮& & ⋮\\ C_{1n}& C_{2n}& \cdots & C_{nn}\end{array}\right)$. (3.15)

Using this definition together with the properties of the determinant, the following extension of Theorem 2 provides an explicit formula for the inverse of an $n\times n$ matrix.

Theorem 7.  A square matrix $A$ is invertible if and only if $\det \left(A\right)\ne 0$, in which case the inverse is given by

$A^{-1}={\displaystyle \frac{1}{\det \left(A\right)}}\text{adj(}A\text{)}$ (3.16)

Equation (3.16) in the previous theorem is then used to find an explicit formula for the solution of a system of $n$ linear equations with $n$ unknowns when the coefficient matrix $A$ is invertible. Such a formula is called Cramer’s Rule and is summarized in the following theorem:

Theorem 8.  Cramer’s Rule

If $A\overrightarrow{x}=\overrightarrow{b}$ is a system of $n$ linear equations with $n$ unknowns and $\det \left(A\right)\ne 0$ (i.e., $A$ is invertible), then the system has a unique solution given by

$x_1={\displaystyle \frac{\det \left(A_1\right)}{\det \left(A\right)}}$;   $x_2={\displaystyle \frac{\det \left(A_2\right)}{\det \left(A\right)}}$;   $\dots x_n={\displaystyle \frac{\det \left(A_n\right)}{\det \left(A\right)}}$,

where $A_j$ is the matrix obtained by replacing the entries in the $jth$ column of $A$ by the entries in

$\overrightarrow{b}=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right)$.

Exercises

Work on the following textbook exercises from “Exercise Set 2.3” (pp. 127-128):

• 19, 23, 25, 27, 29, 31; and
• “True-False Exercises” (e, f, j, k, l).

3.5  Applications of Linear Algebra to Geometry: Constructing Curves and Surfaces Using Determinants

This section illustrates a direct application of the concept of the determinant of a matrix to geometry. In particular, it provides a procedure to construct curves and surfaces using part of the Equivalent Statements Theorem 6 stated in Section 3.3.

Reading Assignment

Read, and study, pages 528-532 of the textbook (to “Exercise Set 10.1”). The reading starts by stating a theorem that is a result of parts $\left(b\right)$ and $\left(g\right)$ of the Equivalent Statements Theorem 6. Specifically, if we consider only these parts of the Equivalent Statements Theorem we can conclude the following:

Theorem 9.  A homogeneous system with as many equations as unknowns has only the trivial solution if and only if the determinant of the coefficient matrix is not equal to zero.

The textbook authors simply rewrite this theorem in its alternative form as:

Theorem 10.  A homogeneous system with as many equations as unknowns has a nontrivial solution if and only if the determinant of the coefficient matrix is equal to zero.

This last theorem is then used to find the equations in ${ℝ}^2$ of (a) a line that passes through two points; (b) a circle that passes through three points; and (c) a general conic that passes through five points. Also, a really interesting application of how to find the equation of an orbit of an asteroid about the sun is provided in the reading.

The reading goes even further by using the theorem to find the equations in $R^3$ of (a) the plane that passes through three noncollinear points; and (b) the sphere that passes through four points.

Exercises

Work on the following textbook exercises from “Exercise Set 10.1” (pp. 533-533):

• 1, 2(a), 3, 4(a), 6(a), 8, 11, and 13.