Wize University Linear Algebra Textbook > Determinants

Basics of Determinants

Basics of Determinants

Determinant Definition

Every square matrix has an associated special scalar number called the determinant of the matrix.

The determinant has many uses, including calculating inverse matrices and finding solutions to SLEs.

Notation

Given a square matrix $A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n}\\ a_{21} & a_{22} & \dots & a_{2n}\\ \vdots & \vdots & & \vdots\\ a_{n1} & a_{n2} & \dots & a_{nn}\\ \end{bmatrix}$ , the determinant of $A$ is denoted with vertical bars:

$\detp{A} = \left \vert \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1n}\\ a_{21} & a_{22} & \dots & a_{2n}\\ \vdots & \vdots & & \vdots\\ a_{n1} & a_{n2} & \dots & a_{nn}\\ \end{array} \right\vert$

Calculating Determinants of Small Matrices

$1\times1$ matrix: $\detp{[\ a\ ]}=a$
$2\times2$ matrix: ${\rm det} \begin{bmatrix} a&b\\ c&d \end{bmatrix} \ =\ \left \vert \begin{array}{rr} a&b\\ c&d \end{array} \right \vert \ =\ ad-bc$

Determinant Properties

Let $A_{n \times n}$ and $\ B_{n \times n}$ be square matrices.

If $A$ has a zero row or zero column, then $\detp{A}=0$
If two rows or two columns are scalar multiples of one another, then $\detp{A}=0$
$\detp{A}=\detp{A^T}$
$\detp{A^{-1}}=\dfrac{1}{\detp{A}}$
$\detp{cA}=c^n\ \detp{A}$
$\detp{AB} = \detp{A} \cdot \detp{B}$

Note: In general, $\detp{A+B} \colorTwo{\neq} \detp{A} + \detp{B}$ .

Triangular Matrices

Upper triangular matrix: all entries under the main diagonal are 0s. Example: $\begin{bmatrix} \colorTwo a & \colorTwo b & \colorTwo c\\ 0 & \colorTwo d & \colorTwo e\\ 0 & 0 & \colorTwo f\\ \end{bmatrix}$
Lower triangular matrix: all entries above the main diagonal are 0s. Example: $\begin{bmatrix} \colorTwo a & 0& 0 \\ \colorTwo b & \colorTwo c &0\\ \colorTwo d & \colorTwo e & \colorTwo f\\ \end{bmatrix}$
Diagonal matrix: both upper and lower triangular. All entries that are not on the main diagonal are 0s. Example: $\begin{bmatrix} \colorTwo a & 0 & 0\\ 0 & \colorTwo b &0\\ 0 & 0 & \colorTwo c\\ \end{bmatrix}$

Example: Basics of Determinants

Part A)

Find the determinant of $A=\begin{bmatrix} -1&2\\ -3&4 \end{bmatrix}$ .

${\rm det}(A) \ =\ ad - bc \ =\ \left(−1\right)\left(4\right)−\left(2\right)\left(-3\right) \ =\ −4−\left(−6\right) \ =\ 2$

Part B)

Find the determinant of $B= \left[\begin{array}{rrr} 3&0&0\\ 7&-5&0\\ 6&2&-1 \end{array}\right]$ .

Notice that $B$ is lower triangular (0s above the main diagonal).

Then its determinant is the product of the diagonal entries.

$\detp{B} \ =\ (3)(-5)(-1) \ =\ 15$

Part C)

Given $C= \left[ \begin{array}{rr} 2&3\\ -1&1 \end{array} \right]$ and $D= \left[ \begin{array}{rr} -3&3\\ 1&-1 \end{array} \right]$ , find $\detp{3CD}$ without using matrix multiplication.

Using determinant properties:

$\detp{3CD} = 3^2 \cdot\detp{CD}$
Since the product $CD$ is a $2 \times 2$ matrix, we must raise the scalar to the power of $2$ .
$\detp{CD} = \detp{C} \ \detp{D}$

So we can compute:

$\begin{aligned} \detp{3CD} &= 3^2\cdot \detp{C}\cdot\detp{D}\\[0.5em] &=9 \cdot \left\vert \begin{array}{rr} 2&3\\ -1&1 \end{array} \right\vert \cdot \left\vert \begin{array}{rr} -3&3\\ 1&-1 \end{array} \right\vert \\[1em] &=9 \cdot \big[(2)(1)-(3)(-1)\big]\cdot \big[(-3)(-1)-(3)(1)\big] \\[0.5em] &=9 \cdot 5 \cdot 0 \\[0.5em] &= 0 \end{aligned}$

Note: $\detp{D}=0$ since it contains rows (and columns) that are scalar multiples of one another.

Let $C$ be an invertible matrix with $\det C = 3$ , and let $B= \left[ \begin{array}{rrr} -2 & -1 & 0\\ 0 & 1 & 0\\ 0 & 0 & -5\\ \end{array} \right]$ .

If $A = C^{-1}BC$ , calculate $\det A$ .

I don't know

Answer the following true/false questions.

True or False?

Given $A_{4 \times 4}$ and $\det A = -3$ , then $\detp{A^T}=\dfrac{1}{\detp{A^{-1}}}$ .

Your answer

I don't know

Extra Practice

Determinants: Basics

Find the determinant of the matrix $\begin{bmatrix}2&-4\\ 1&-3\end{bmatrix}$ .

Determinants: Basics

$A= \begin{bmatrix} 4&5\\ -1&2 \end{bmatrix}, \ B= \begin{bmatrix} -1&0&0&0\\ 2&1&0&0\\ 30&20&10&0\\ 1&3&-4&-10 \end{bmatrix}, \ C= \begin{bmatrix} -1&5\\ 5&-25 \end{bmatrix}, \ I_{10}=10\times 10\ \text{ identity matrix}$

Compute $det(AC)-det(10B^T)+det(2I_{10})$ .

Determinants: Basics

$A$ is a $3\times 3$ matrix where $det\ A=2$ .

Find $det(5A2A^{-1}A^T)$ .

Determinants: Basics

If $A= \begin{bmatrix} 1&2&3&4\\ 0&2&1&-5\\ 0&0&-3&0\\ 0&0&0&-1 \end{bmatrix}$ , calculate $det(Adj\ A)$ .

Basics of Determinants

Find $\text{det}\begin{bmatrix} 3&2&-4&1\\ 0&1&-1&2\\ -1&-7&5&0\\ -6&-4&8&-2 \end{bmatrix}$ .

Determinant Properties Practice Question

If 𝑨 = 𝑪-1𝟏𝑩𝑪, where C is an invertible matrix, $det(𝑩) = 12; det(𝑪) = 3. \text{Calculate}\ \ det(𝑨).$ and

Practice: Determinant Property

Practice Question: Determinant Property

Let 𝐶 be an invertible matrix, det (𝐵) = 10, and det (𝐶) = 3.

If 𝐴 = 𝐶−1𝐵𝐶, calculate det (𝐴).

133 - FML 1 - 18.1W e.g. 7

Recall that diagonal matrices are substantially easier to work with than non-diagonal matrices:

e.g. If

$\text{A}=\text{diag}\left(a,b,c\right) = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}$

Determinant Properties Practice Question

If 𝑨 = 𝑪-1𝟏𝑩𝑪, where C is an invertible matrix, $det(𝑩) = 12; det(𝑪) = 3. \text{Calculate}\ \ det(𝑨).$ and

Basics of Determinants

Let $C$ be an invertible matrix with $\det C = 3$ , and let $B= \left[ \begin{array}{rrr} -2 & -1 & 0\\ 0 & 1 & 0\\ 0 & 0 & -5\\ \end{array} \right]$ .

If $A = C^{-1}BC$ , calculate $\det A$ .

Find the determinant of $A=\begin{bmatrix} 0&2&0&3\\ −1&4&0&−1\\ 3&1&1&0\\ 1&0&2&−2 \end{bmatrix}$

Compute the determinant of

$\begin{bmatrix} 3&1&−1&1\\ 1&0&2&3\\ −2&0&3&4\\ 1&0&2&2 \end{bmatrix}$

Basics of Determinants

Answer the following true/false questions.

133 - FML 3 - 18.1W e.g. 71

$\bcb{\boldsymbol{ \ul{B} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 1 \\ 2 & x & 2 \end{bmatrix} }}$ such that $\bcb{\boldsymbol{ \detp {\ul{B}} = -8 }}$ , then $\bcb{\boldsymbol{ x }}$ is:

\text{(a)}\;\boldsymbol{ \frac{1}{11} } \qquad\qquad\qquad \text{(b)}\; \boldsymbol{ -1}\qquad\qquad\qquad \text{(c)}\; \boldsymbol{ 0}\qquad\qquad\qquad \text{(d)}\; \text{\bf{None of the above}}

133 - FML 3 - 18.1W e.g. 11

Find all possible values of $\bcb{\boldsymbol{ \detp{\ul{A}} }}$ given that $\bcb{\boldsymbol{ adj(A) = \begin{bmatrix} 1 & 2 & 4 \\ -1 & 2 & 2 \\ 0 & 1 & 8 \end{bmatrix} }}$

133 - FML 3 - 18.1W e.g. 17.3

If $\bcb{\boldsymbol{ \ul{A} }}$ is a $\bcb{\boldsymbol{ 3 \times 3}}$ matrix with $\bcb{\boldsymbol{ \detp{\ul{A}} = -4}}$ , find:

\boldsymbol{ \detp{-\mtran{A}} }

Basics of Determinants

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