Math 1410, Spring 2020

Determinant Properties and Applications — Pandemic Lockdown Style

Sean Fitzpatrick
University of Lethbridge

Recap

Warm-Up

Compute the determinant of the given matrices, possibly after doing a row operation:

  1. \(A = \bbm 1\amp 4\amp 0\\2\amp -3\amp 5\\0\amp -1\amp -2\ebm\)

  2. \(B = \bbm 3\amp 0 \amp -2\amp 1\\ -2\amp 1 \amp 1 \amp 3\\ 0 \amp -1\amp 2 \amp -3\\ 4\amp 0\amp 1\amp 0\ebm\)

Properties of Determinants

Effect of row operations

Theorem:

  1. If \(B\) is obtained from \(A\) using the row operation \(R_i\leftrightarrow R_j\text{,}\) then \(\det B = -\det A\text{.}\)

  2. If \(B\) is obtained from \(A\) using the row operation \(kR_i \to R_i\text{,}\) then \(\det B = k\det A\text{.}\)

  3. If \(B\) is obtained from \(A\) using the row operation \(R_i+kR_j\to R_i\text{,}\) then \(\det B=\det A\text{.}\)

Note: these effects are most easily observed in elementary matrices!

Examples

  1. Suppose \(B\) is obtained from \(A\) using the folloing row operations:

    1. \(\frac{1}{4}R_1\to R_1\)

    2. \(R_2-4R_1\to R_2\)

    3. \(R_2\leftrightarrow R_3\)

    4. \(R_3+3R_2\to R_3\)

    If \(\det B = -7\text{,}\) what is \(\det A\text{?}\)

  2. If \(A\) is a \(4\times 4\) matrix and \(\det A = -3\text{,}\) what is the value of \(\det(2A)\text{?}\)

Properties of Determinants

Theorem:

Let \(A\) and \(B\) be \(n\times n\) matrices. Then:

  1. \(\det(AB) = \det(A)\det(B)\)

  2. \(\det(A^T) = \det(A)\)

  3. \(\det(kA)=k^n\det(A)\)

Theorem:

A matrix \(A\) is invertible if and only if \(\det(A)\neq 0\text{.}\) Furthermore, if \(A\) is invertible, then

\begin{equation*} \det(A^{-1})=\frac{1}{\det(A)}\text{.} \end{equation*}

Examples

Given that \(\det A = 3\) and \(\det B = -2\text{,}\) what is the value of:

  1. \(\det(A^2B^3)\)

  2. \(\det(B^{-1}AB)\)

  3. \(\det(2AB^{-1})\)

More examples

What can you say about \(\det A\) if:

  1. \(A^2=A\)

  2. \(A^4=I\)

  3. \(PA=P\text{,}\) where \(P\) is invertible.

The adjugate formula for the inverse

The cofactor matrix

Recall: given an \(n\times n\) matrix \(A\text{,}\) the \((i,j)\) cofactor is the number \(C_{ij}=(-1)^{i+j}\det M_{ij}\text{,}\) where \(M_{ij}\) is the \((i,j)\) minor.

The matrix of cofactors of \(A\) is the matrix \(\cof(A)\) whose \((i,j)\) entry is \(C_{ij}\text{.}\)

Example: find \(\cof(A)\) if \(A = \bbm 2\amp -1\amp 3\\0\amp 4\amp -2\\1\amp -1\amp 0\ebm\text{.}\)

The adjugate matrix

Definition ():

The adjugate of an \(n\times n\) matrix \(A\) is given by \(\adj(A)=\cof(A)^T\text{.}\)

Theorem:

For any \(n\times n\) matrix \(A\text{,}\)

\begin{equation*} A\cdot \adj(A) = \det(A)I_n\text{.} \end{equation*}

Examples

Use the formula \(\di A^{-1}=\frac{1}{\abs{A}}\adj(A)\) to compute the inverse of:

  1. \(A = \bbm 2\amp 1\amp -3\\3\amp 0\amp 2\\0\amp 1\amp 4\ebm\)

  2. \(A = \bbm 1\amp 0\amp x\\0\amp -x\amp 2\\x\amp 0\amp 3\ebm\text{.}\)

Cramer's Rule

Suppose we have a system of \(n\) equations in \(n\) unknowns, written as \(A\vec{x}=\vec{b}\text{.}\)

If \(\det A =0\text{,}\) then \(A\) is not invertible, and this system has either no solution, or infinitely many solutions.

If \(\det A\neq 0\text{,}\) then

\begin{equation*} \vec{x} = A^{-1}\vec{b} = \frac{1}{\abs{A}}\adj(A)\vec{b}\text{.} \end{equation*}

Result: if \(A_i\) denotes the matrix obtained by replacing column \(i\) of \(A\) by \(\vec{b}\text{,}\) then

\begin{equation*} x_i = \frac{\det{A_i}}{\det{A}}\text{,} \end{equation*}
for \(i=1,2,\ldots, n\text{.}\) (Theoretically and historically interesting, but not very practical.)

Example

Use Cramer's rule to solve the system:

\begin{align*} (\cos\theta) x - (\sin\theta) y \amp = 4 \\ (\sin\theta) x + (\cos\theta) y \amp = W\text{,} \end{align*}
where \(\theta\) is an angle and \(W\) is some unknown (but presumably very important) number.