Determinants and Matrix Inverses

Core Concepts

Compute determinants, invert matrices, and solve matrix equations of the form AX = B.

Determinants and Matrix Inverses

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What is the formula for the determinant of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

$$ \det(A) = ad - bc $$

What condition involving the determinant dictates whether a square matrix $A$ has a multiplicative inverse $A^{-1}$?

A square matrix $A$ has an inverse if and only if $\det(A) \neq 0$. If $\det(A) = 0$, the matrix is called singular and cannot be inverted.

What is the formula for the inverse of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

$$ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} $$ (valid when $ad - bc \neq 0$).

Given the matrix equation $A\mathbf{x} = \mathbf{b}$, where $A$ is invertible, what is the exact algebraic solution for $\mathbf{x}$?

$$ \mathbf{x} = A^{-1}\mathbf{b} $$ Note: Because matrix multiplication is not commutative, $\mathbf{x} \neq \mathbf{b}A^{-1}$. You must left-multiply both sides by $A^{-1}$.

Evaluate the determinant: $$ \begin{vmatrix} 3 & -2 \\ 4 & 5 \end{vmatrix} $$

$$ \det = (3)(5) - (-2)(4) = 15 - (-8) = 23 $$

How do you evaluate a $3 \times 3$ determinant using cofactor expansion along the first row?

For $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$: $$ \det(A) = a\begin{vmatrix} e & f \\ h & i \end{vmatrix} - b\begin{vmatrix} d & f \\ g & i \end{vmatrix} + c\begin{vmatrix} d & e \\ g & h \end{vmatrix} $$ Alternating signs $(+,-,+)$ must be applied to the entries of the row.

Solve the system using an inverse matrix: $$ \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ 11 \end{bmatrix} $$

1. $\det(A) = (2)(3) - (1)(5) = 1$. 2. $A^{-1} = \frac{1}{1} \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$. 3. Multiply: $$\mathbf{x} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix} \begin{bmatrix} 4 \\ 11 \end{bmatrix} = \begin{bmatrix} 12 - 11 \\ -20 + 22 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$$ Solution: $x = 1$, $y = 2$.

What is the determinant of an upper or lower triangular matrix?

The determinant of any triangular matrix is simply the product of the entries along its main diagonal.

Determinants and Matrix Inverses

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