Determinants and Inverses of Nonsingular Matrices

Combining results of Theorem th:detofsingularmatrix and Theorem th:nonsingularequivalency1 shows that the following statements about matrix \(A\) are equivalent:

  • \(A^{-1}\) exists
  • Any equation \(Ax=\vec {b}\) has a unique solution
  • \(\det {A}\neq 0\)

In this section we will take a closer look at the relationship between the determinant of a nonsingular matrix \(A\), solution to the system \(A\vec {x}=\vec {b}\), and the inverse of \(A\).

Cramer’s Rule

We begin by establishing a formula that allows us to express the unique solution to the system \(A\vec {x}=\vec {b}\) in terms of the determinant of \(A\), for a nonsingular matrix \(A\). This formula is called Cramer’s rule.

Consider the system

\[\begin{array}{ccccc} ax& +&by&=&e\\ cx & +&dy&= &f \end{array}\]

The system can be written as a matrix equation

\[\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}e\\f\end{bmatrix}\]

Using one of our standard methods for solving systems we find that

\[x=\frac {ed-bf}{ad-bc}\quad \text {and}\quad y=\frac {af-ec}{ad-bc}\]

Observe that the denominators in the expressions for \(x\) and \(y\) are the same and equal to \(\det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}\).

A close examination shows that the numerators of expressions for \(x\) and \(y\) can also be interpreted as determinants of matrices. The numerator of the expression for \(x\) is the determinant of the matrix that is formed by replacing the first column of \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) with \(\begin{bmatrix}e\\f\end{bmatrix}\). The numerator of the expression for \(y\) is the determinant of the matrix that is formed by replacing the second column of \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) with \(\begin{bmatrix}e\\f\end{bmatrix}\). Thus, \(x\) and \(y\) can be written as

\[x=\frac {ed-bf}{ad-bc}=\frac {\begin{vmatrix}e&b\\f&d\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}\quad \text {and}\quad y=\frac {af-ec}{ad-bc}=\frac {\begin{vmatrix}a&e\\c&f\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}\]

Note that a unique solution to the system exists if and only if the determinant of the coefficient matrix is not zero.

It turns out that a solution to any square system \(A\vec {x}=\vec {b}\) can be expressed using ratios of determinants, provided that \(A\) is nonsingular. The general formula for the \(i^{th}\) component of the solution vector is

\[x_i=\frac {\det {(\text {matrix } A \text { with column } i \text { replaced by } \vec {b})}}{\det {A}}\]

To formalize this expression, we need to introduce some notation. Given a matrix

\[A=\begin{bmatrix} | & |& &|\\ \vec {a}_1 & \vec {a}_2&\dots &\vec {a}_n\\ | & |& &| \end{bmatrix}\]

and a vector \(\vec {b}\) we use

\[A_i(\vec {b})\]

to denote the matrix obtained from \(A\) by replacing the \(i^{th}\) column of \(A\) with \(\vec {b}\). In other words,

\begin{equation}\label {eq:AiNotation} A_i(\vec {b})=\begin{bmatrix} | & |& &|&|&|&&|\\ \vec {a}_1 & \vec {a}_2&\dots &\vec {a}_{i-1}&\vec {b}&\vec {a}_{i+1}&\dots &\vec {a}_n\\ | & |& &|&|&|&&| \end{bmatrix} \end{equation}

Using our new notation, we can write the \(i^{th}\) component of the solution vector as

\[x_i=\frac {\det {A_i(\vec {b})}}{\det {A}}\]

We will work through a couple of examples before proving this result as a theorem.

We are now ready to state and prove Cramer’s rule as a theorem.

For this proof we will need to think of matrices in terms of their columns. Thus,

\[A=\begin{bmatrix} | & |& &|\\ \vec {a}_1 & \vec {a}_2&\dots& \vec {a}_n\\ | & |& &| \end{bmatrix}\]
We will also need the identity matrix \(I\). The columns of \(I\) are standard unit vectors.
\[I=\begin{bmatrix} | & |& &|\\ \vec {e}_1 & \vec {e}_2&\dots& \vec {e}_n\\ | & |& &| \end{bmatrix}\]
Recall that
\[A_i(\vec {b})=\begin{bmatrix} | & |& &|&|&|&&|\\ \vec {a}_1 & \vec {a}_2&\dots &\vec {a}_{i-1}&\vec {b}&\vec {a}_{i+1}&\dots& \vec {a}_n\\ | & |& &|&|&|&&| \end{bmatrix}\]

Similarly,

\[I_i(\vec {x})=\begin{bmatrix} | & |& &|&|&|&&|\\ \vec {e}_1 & \vec {e}_2&\dots &\vec {e}_{i-1}&\vec {x}&\vec {e}_{i+1}&\dots& \vec {e}_n\\ | & |& &|&|&|&&| \end{bmatrix}\]
Observe that \(x_i\) is the only non-zero entry in the \(i^{th}\) row of \(I_i(\vec {x})\). Cofactor expansion along the \(i^{th}\) row gives us
\begin{equation}\label {eq:cramerix}\det {I_i(\vec {x})}=x_i\end{equation}
Now, consider the product \(A\Big (I_i(\vec {x})\Big )\)
\begin{align*}A\Big (I_i(\vec {x})\Big )&=\begin{bmatrix} | & |& &|\\ \vec {a}_1 & \vec {a}_2&\dots &\vec {a}_n\\ | & |& &| \end{bmatrix}\begin{bmatrix} | & |& &|&|&|&&|\\ \vec {e}_1 & \vec {e}_2&\dots &\vec {e}_{i-1}&\vec {x}&\vec {e}_{i+1}&\dots &\vec {e}_n\\ | & |& &|&|&|&&| \end{bmatrix}\\ &=\begin{bmatrix} | & |& &|&|&|&&|\\ \vec {a}_1 & \vec {a}_2&\dots &\vec {a}_{i-1}&A\vec {x}&\vec {a}_{i+1}&\dots &\vec {a}_n\\ | & |& &|&|&|&&| \end{bmatrix}\\ &=\begin{bmatrix} | & |& &|&|&|&&|\\ \vec {a}_1 & \vec {a}_2&\dots &\vec {a}_{i-1}&\vec {b}&\vec {a}_{i+1}&\dots &\vec {a}_n\\ | & |& &|&|&|&&| \end{bmatrix}=A_i(\vec {b}) \end{align*}

This gives us

\[AI_i(\vec {x})=A_i(\vec {b})\]
\[\det {AI_i(\vec {x})}=\det {A_i(\vec {b})}\]
\[\det {A}\det {I_i(\vec {x})}=\det {A_i(\vec {b})}\]
By our earlier observation in (2) we have
\[\det {A}x_i=\det {A_i(\vec {b})}\]
\(A\) is nonsingular, so \(\det {A}\neq 0\). Thus
\[x_i=\frac {\det {A_i(\vec {b})}}{\det {A}}\]

Finding the determinant is computationally expensive. Because Cramer’s rule requires finding many determinants, it is not a computationally efficient way of solving a system of equations. However, Cramer’s rule is often used for small systems in applications that arise in economics, natural, and social sciences, particularly when solving for only a subset of the variables.

Adjugate Formula for the Inverse of a Matrix

In Practice Problem ?? we used the row reduction algorithm to show that if

\[A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\]

is nonsingular then

\begin{equation}\label {eq:twobytwoinverse}A^{-1}=\frac {1}{\det {A}}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\end{equation}

This formula is a special case of a general formula for the inverse of a nonsingular square matrix. Just like the formula for a \(2\times 2\) matrix, the general formula includes the coefficient \(\frac {1}{\det {A}}\) and a matrix related to the original matrix. We will now derive the general formula using Cramer’s rule.

Let \(A\) be an \(n\times n\) nonsingular matrix. When looking for the inverse of \(A\), we look for a matrix \(X\) such that \(AX=I\). We will think of matrices in terms of their columns

\[I=\begin{bmatrix} | & |& &|\\ \vec {e}_1 & \vec {e}_2&\dots& \vec {e}_n\\ | & |& &| \end{bmatrix}\quad \text {and}\quad X=\begin{bmatrix} | & |& &|\\ \vec {x}_1 & \vec {x}_2&\dots& \vec {x}_n\\ | & |& &| \end{bmatrix}\]

If \(AX=I\) then we must have

\[A\vec {x}_1=\vec {e}_1\]
\[A\vec {x}_2=\vec {e}_2\]
\[\vdots \]
\[A\vec {x}_n=\vec {e}_n\]

This gives us \(n\) systems of equations. Solution vectors to these systems are the columns of \(X\). Thus, the \(j^{th}\) column of \(X\) is

\[\vec {x}_j=\begin{bmatrix}x_{1j}\\x_{2j}\\\vdots \\x_{nj}\end{bmatrix}\quad \text {such that}\quad A\vec {x}_j=\vec {e}_j\]

By Cramer’s rule

\[x_{ij}=\frac {\det {A_i(\vec {e}_j)}}{\det {A}}\]

But

\[A_i(\vec {e})=\begin{bmatrix} | & |& &|&|&|&&|\\ \vec {a}_1 & \vec {a}_2&\dots &\vec {a}_{i-1}&\vec {e}_j&\vec {a}_{i+1}&\dots& \vec {a}_n\\ | & |& &|&|&|&&| \end{bmatrix}\]

To find \(\det {A_i(\vec {e}_j)}\), we can expand along the \(i^{th}\) column of \(A_i(\vec {e}_j)\). But the \(i^{th}\) column of \(A_i(\vec {e}_j)\) is the vector \(\vec {e}_j\) which has 1 in the \(j^{th}\) spot and zeros everywhere else. Thus

\[\det {A_i(\vec {e}_j)}=(-1)^{i+j}\det {A_{ji}}=C_{ji}\]

We now have

\[\vec {x}_j=\begin{bmatrix}C_{j1}/\det {A}\\C_{j2}/\det {A}\\\vdots \\C_{jn}/\det {A}\end{bmatrix}=\frac {1}{\det {A}}\begin{bmatrix}C_{j1}\\C_{j2}\\\vdots \\C_{jn}\end{bmatrix}\]

Thus,

\[A^{-1}=X=\frac {1}{\det {A}}\begin{bmatrix}C_{11}&C_{21}&\ldots&C _{n1}\\C_{12}&C_{22}&\ldots&C _{n2}\\\vdots& \vdots& \ddots& \vdots \\ C_{1n}&C_{2n}&\ldots&C _{nn}\end{bmatrix}\]

The matrix of cofactors of \(A\) is called the adjugate of \(A\). We write

\[\text {adj}(A)=\begin{bmatrix}C_{11}&C_{21}&\ldots&C _{n1}\\C_{12}&C_{22}&\ldots&C _{n2}\\\vdots& \vdots& \ddots& \vdots \\ C_{1n}&C_{2n}&\ldots&C _{nn}\end{bmatrix}\]

We summarize our result as a theorem

Practice Problems

Problems -

Use Cramer’s rule to solve each of the following systems.

\[\begin{array}{ccccc} -3x& +&2y&=&1\\ 2x & -&y&= &4 \end{array}\]
Answer:
\[x=\answer {9}\quad y=\answer {14}\]
\[\begin{bmatrix}2&-5&1\\6&0&-2\\-3&1&1\end{bmatrix}\vec {x}=\begin{bmatrix}1\\4\\3\end{bmatrix}\]
Answer:
\[\vec {x}=\begin{bmatrix}\answer {28/5}\\\answer {5}\\\answer {74/5}\end{bmatrix}\]
Consider the equation
\[\begin{bmatrix}1&1&1&1\\2&3&2&-1\\3&-2&3&-2\\2&1&1&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=\begin{bmatrix}10\\10\\0\\11\end{bmatrix}\]
1.
Solve for \(x_2\) using Cramer’s Rule.

Answer:

\[x_2=\answer {2}\]
2.
If you had to solve for all four variables, which method would you use? Why?

Problems -

Use Theorem 8 to find the inverse of each of the following matrices.

\[A=\begin{bmatrix}2&7\\1&3\end{bmatrix}\]
Answer:
\[A^{-1}=\begin{bmatrix}\answer {-3}&\answer {7}\\\answer {1}&\answer {-2}\end{bmatrix}\]
\[A=\begin{bmatrix}2&1&4\\4&-2&1\\0&3&-1\end{bmatrix}\]
Answer:
\[A^{-1}=\frac {1}{\answer {50}}\begin{bmatrix}\answer {-1}&\answer {13}&\answer {9}\\\answer {4}&\answer {-2}&\answer {14}\\\answer {12}&\answer {-6}&\answer {-8}\end{bmatrix}\]
Show that the formula in (3) is a special case of the formula in Theorem 8 by showing that
\[\mbox {adj}\left (\begin{bmatrix}a&b\\c&d\end{bmatrix}\right )=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\]