This is an old revision of the document!
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/action.php on line 14
Table of Contents
Notation
To save having to write $\det$ all the time, we sometimes write the entries of a matrix inside vertical bars $|\ |$ to mean the determinant of that matrix. Using this notation (and doing a few steps in our heads), we can rewrite the previous example as:
\begin{align*}\def\vm#1{\begin{vmatrix}#1\end{vmatrix}}\vm{1&2&3\\7&8&9\\11&12&13} &= 1\vm{8&9\\12&13} -2\vm{7&9\\11&13} + 3\vm{7&8\\11&12}\\ &=-4 -2(-8)+3(-4)\\ &=0.\end{align*}
Step 4: the determinant of an $n\times n$ matrix
Definition
If $\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}A=\mat{a_{11}&a_{12}&\dots&a_{1n}\\\vdots&&&\vdots\\a_{n1}&a_{n2}&\dots&a_{nn}}$ is an $n\times n$ matrix, then \[\det A=a_{11}C_{11}+a_{12}C_{12}+\dots+a_{1n}C_{1n}.\] Here $C_{ij}$ are the cofactors of $A$.
This formula is called the Laplace expansion of $\det A$ along the first row, since $a_{11}, a_{12},\dots,a_{1n}$ make up the first row of $A$.
Example
\begin{align*} \def\vm#1{\begin{vmatrix}#1\end{vmatrix}} \vm{\color{red}1&\color{red}0&\color{red}2&\color{red}3\\0&2&1&-1\\2&0&0&1\\3&0&4&2} &= \color{red}1\vm{\color{blue}2&\color{blue}1&\color{blue}-1\\0&0&1\\0&4&2}-\color{red}0\vm{0&1&-1\\2&0&1\\3&4&2}+\color{red}2\vm{\color{orange}0&\color{orange}2&\color{orange}{-1}\\2&0&1\\3&0&2}-\color{red}3\vm{\color{purple}0&\color{purple}2&\color{purple}1\\2&0&0\\3&0&4}\\ &= 1\left(\color{blue}2\vm{0&1\\4&2}-\color{blue}1\vm{0&1\\0&2}\color{blue}{-1}\vm{0&0\\0&4}\right)-0+2\left(\color{orange}0-\color{orange}{2}\vm{2&1\\3&2}\color{orange}{-1}\vm{2&0\\3&0}\right)-3\left(\color{purple}0-\color{purple}2\vm{2&0\\3&4}+\color{purple}1\vm{2&0\\3&0}\right)\\ &=1(2(-4)-0-0)+2(-2(1)-0)-3(-2(8)+0)\\ &=-8-4+48\\ &=36. \end{align*}
Theorem: Laplace expansion along any row or column gives the determinant
- For any fixed $i$: $\det(A)=a_{i1}C_{i1}+a_{i2}C_{i2}+\dots+a_{in}C_{in}$ (Laplace expansion along row $i$)
- For any fixed $j$: $\det(A)=a_{1j}C_{1j}+a_{2j}C_{2j}+\dots+a_{nj}C_{nj}$ (Laplace expansion along column $j$)
Example
We can make life easier by choosing expansion rows or columns with lots of zeros, if possible. Let's redo the previous example with this in mind:
\begin{align*} \def\vm#1{\begin{vmatrix}#1\end{vmatrix}} \vm{1&\color{red}0&2&3\\0&\color{red}2&1&-1\\2&\color{red}0&0&1\\3&\color{red}0&4&2} &= -\color{red}0+\color{red}2\vm{1&2&3\\\color{purple}2&\color{purple}0&\color{purple}1\\3&4&2}-\color{red}0+\color{red}0\\ &=2\left(-\color{purple}2\vm{2&3\\4&2}+\color{purple}0-\color{purple}1\vm{1&2\\3&4}\right)\\ &=2(-2(-8)-(-2))\\ &=36. \end{align*}
Definition: upper triangular matrices
An $n\times n$ matrix $A$ is upper triangular if all the entries below the main diagonal are zero.
Definition: diagonal matrices
An $n\times n$ matrix $A$ is diagonal if the only non-zero entries are on its main diagonal.
Corollary: the determinant of upper triangular matrices and diagonal matrices
- The determinant of an upper triangular $n\times n$ matrix is the product of its diagonal entries: $\det(A)=a_{11}a_{22}\dots a_{nn}$.
- The determinant of an $n\times n$ diagonal matrix is the product of its diagonal entries: $\det(A)=a_{11}a_{22}\dots a_{nn}$.
Proof
- This is true for $n=1$, trivially. For $n>1$, assume inductively that it is true for $(n-1)\times (n-1)$ matrices and use the Laplace expansion of an upper triangular $n\times n$ matrix $A$ along the first column of $A$ to see that $\det(A)=a_{11}C_{11}+0+\dots+0=a_{11}C_{11}$. Now $C_{11}$ is the determinant of the $(n-1)\times (n-1)$ matrix formed by removing the first row and and column of $A$, and this matrix is upper triangular with diagonal entries $a_{22},a_{33},\dots,a_{nn}$. By our inductive assumption, we have $C_{11}=a_{22}a_{33}\dots a_{nn}$. So $\det(A)=a_{11}C_{11}=a_{11}a_{22}a_{33}\dots a_{nn}$ as desired.
- Any diagonal matrix is upper triangular, so this is a special case of statement 1. ■
Examples
- For any $n$, we have $\det(I_n)=1\cdot 1\cdots 1 = 1$.
- For any $n$, we have $\det(5I_n)=5^n$.
- $\def\vm#1{\begin{vmatrix}#1\end{vmatrix}}\vm{1&9&43&23434&4&132\\0&3&43&2&-1423&-12\\0&0&7&19&23&132\\0&0&0&2&0&0\\0&0&0&0&-1&-903\\0&0&0&0&0&6}=1\cdot3\cdot7\cdot2\cdot(-1)\cdot6 = 252$.
Theorem: important properties of the determinant
Let $A$ be an $n\times n$ matrix.
- $A$ is invertible if and only if $\det(A)\ne0$.
- $\det(A^T)=\det(A)$
- If $B$ is another $n\times n$ matrix, then $\det(AB)=\det(A)\det(B)$
Theorem: row/column operations and determinants
Let $A$ be an $n\times n$ matrix, let $c$ be a scalar and let $i\ne j$.
$A_{Ri\to x}$ means $A$ but with row $i$ replaced by $x$.
- If $i\ne j$, then $\det(A_{Ri\leftrightarrow Rj})=-\det(A)$ (swapping two rows changes the sign of det).
- $\det(A_{Ri\to c Ri}) = c\det(A)$ (scaling one row scales $\det(A)$ in the same way)
- $\det(A_{Ri\to Ri + c Rj}) = \det(A)$ (adding a multiple of one row to another row doesn't change $\det(A)$)
- Also, these properties all hold if you change “row” into “column” throughout.
Corollary
If an $n\times n$ matrix $A$ has two equal rows (or columns), then $\det(A)=0$, and $A$ is not invertible.
Proof
If $A$ has two equal rows, row $i$ and row $j$, then $A=A_{Ri\leftrightarrow Rj}$ So $\det(A)=\det(A_{Ri\leftrightarrow Rj}) = -\det(A)$, so $2\det(A)=0$, so $\det(A)=0$.
If $A$ has two equal columns, then $A^T$ has two equal rows, so $\det(A)=\det(A^T)=0$.
In either case, $\det(A)=0$. So $A$ is not invertible.■
Examples
- Swapping two rows changes the sign, so $\def\vm#1{\begin{vmatrix}#1\end{vmatrix}}\vm{0&0&2\\0&3&0\\4&0&0} = -\vm{4&0&0\\0&3&0\\0&0&2}=-4\cdot 3\cdot 2 = -24$.
- Multiplying a row or a column by a constant multiplies the determinant by that constant, so \begin{align*}\vm{ 2&4&6&10\\5&0&0&-10\\9&0&81&99\\1&2&3&4} &= 2\vm{ 1&2&3&5\\5&0&0&-10\\9&0&81&99\\1&2&3&4} \\&= 2\cdot 5\vm{ 1&2&3&5\\1&0&0&-2\\9&0&81&99\\1&2&3&4}\\&= 2\cdot 5\cdot 9 \vm{ 1&2&3&5\\1&0&0&-2\\1&0&9&11\\1&2&3&4}\\&=2\cdot 5\cdot 9\cdot 2 \vm{ 1&1&3&5\\1&0&0&-2\\1&0&9&11\\1&1&3&4}\\&=2\cdot 5\cdot 9\cdot 2\cdot 3 \vm{ 1&1&1&5\\1&0&0&-2\\1&0&3&11\\1&1&1&4}.\end{align*}
- $\det(A_{R1\to R1-R4})=\det(A)$, so \begin{align*}\vm{ 1&1&1&5\\1&0&0&-2\\1&0&3&11\\1&1&1&4} &=\vm{ 0&0&0&1\\1&0&0&-2\\1&0&3&11\\1&1&1&4}=-1\vm{1&0&0\\1&0&3\\1&1&1}+0\\&=-\vm{0&3\\1&1} = -(-3)=3.\end{align*}
- Hence \begin{align*}\vm{ 2&4&6&10\\5&0&0&-10\\9&0&81&99\\1&2&3&4} &= 2\cdot 5\cdot 9\cdot 2\cdot 3 \vm{ 1&1&1&5\\1&0&0&-2\\1&0&3&11\\1&1&1&4} \\&= 2\cdot 5\cdot 9\cdot 2\cdot 3 \cdot 3 = 1620.\end{align*}
Corollary
If $\def\row{\text{row}}\row_j(A)=c\cdot \row_i(A)$ for some $i\ne j$ and some $c\in \mathbb{R}$, then $\det(A)=0$.
Proof
Note that $\row_i(A)-c \cdot\row_j(A)=0$. So $A_{Ri\to Ri-c\,Rj}$ has a zero row, and by Laplace expansion along this row we obtain $\det(A_{Ri\to Ri-c\,Rj})=0$. So $\det(A)=\det(A_{Ri\to Ri-c\,Rj})=0$.■
The effect of EROs on the determinant
We have now seen the effect of each of the three types of ERO on the determinant of a matrix:
- swapping two rows of the matrix multiplies the determinant by $-1$. By swapping rows repeatedly, we are able to shuffle the rows in an arbitrary fashion, and the determinant will either remain unchanged (if we used an even number of swaps) or be multiplied by $-1$ (if we used an odd number of swaps).
- multiplying one of the rows of the matrix by $c\in \mathbb{R}$ multiplies the determinant by $c$; and
- replacing row $j$ by “row $j$ ${}+{}$ $c\times {}$ (row $i$)”, where $c$ is a non-zero real number and $i\ne j$ does not change the determinant.
Moreover, since $\det(A)=\det(A^T)$, this all applies equally to columns instead of rows.
We can use EROs to put a matrix into upper triangular form, and then finding the determinant is easy: just multiply the diagonal entries together. We just have to keep track of how the determinant is changed by the EROs of types 1 and 2.
Example: using EROs to find the determinant
\begin{align*}\def\vm#1{\begin{vmatrix}#1\end{vmatrix}} \vm{1&3&1&3\\\color{red}4&\color{red}8&\color{red}0&\color{red}{12}\\0&1&3&6\\2&2&1&6}&= \color{red}{4}\vm{1&3&1&\color{blue}3\\1&2&0&\color{blue}3\\0&1&3&\color{blue}6\\2&2&1&\color{blue}6}\\&=4\cdot \color{blue}3\vm{\color{green}1&3&1&1\\\color{red}1&2&0&1\\\color{red}0&1&3&2\\\color{red}2&2&1&2} \\&=12\vm{1&3&1&1\\\color{blue}0&\color{blue}{-1}&\color{blue}{-1}&\color{blue}{0}\\\color{blue}0&\color{blue}1&\color{blue}3&\color{blue}2\\0&-4&-1&-0} \\&=\color{blue}{-}12\vm{1&3&1&1\\0&\color{green}1&3&2\\0&\color{red}{-1}&{-1}&{0}\\0&\color{red}{-4}&-1&0} \\&=-12\vm{1&3&1&1\\0&1&3&2\\0&0&\color{green}2&2\\0&0&\color{red}{11}&8} \\&=-12\vm{1&3&1&1\\0&1&3&2\\0&0&2&2\\0&0&0&-3} \\&=-12(1)(1)(2)(-3)=72. \end{align*}