Determinants

1. Determinants, Minors, and Cofactors

Every square matrix has a special numerical value associated with it called a Determinant, denoted by $\det(A)$ or $|A|$. The determinant tells us immediately if a matrix is invertible.

For a $2 \times 2$ Matrix

If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $\det(A) = |A| = ad - bc$.

The Rule: A matrix is invertible if and only if $\det(A) \neq 0$.

Minors and Cofactors (For $3 \times 3$ and larger)

To find the determinant of larger matrices, we must first break them down into smaller pieces called Minors and Cofactors.

The Minor ($M_{ij}$): Let $A$ be an $n \times n$ matrix. The minor of the entry $a_{ij}$ is the determinant of the sub-matrix that remains after you physically cross out the $i$-th row and $j$-th column.

The Cofactor ($C_{ij}$): The cofactor is simply the Minor, but with a specific positive or negative sign attached to it based on its position in the matrix.

Pro-Tip: You don't actually need to calculate $(-1)^{i+j}$ every time. The signs always follow a "checkerboard" pattern starting with a positive in the top left:

2. Determinants by Cofactor Expansion

To calculate the full determinant of a $3 \times 3$ (or larger) matrix, we use a process called Cofactor Expansion.

The Theorem

You can pick ANY single row or ANY single column in the matrix. Multiply each entry in that row/column by its corresponding Cofactor, and add the products together. The sum will always be the determinant!

(Expansion along row 1)

The "Save Time" Trick

Since you multiply the entries by their cofactors, always look for a row or column that has the most zeros! If an entry is $0$, you don't even have to calculate its cofactor, because $0 \times \text{Cofactor} = 0$.

Corollary: If a matrix has an entire row of zeros, or an entire column of zeros, its determinant is automatically $0$ (and therefore, it is not invertible).

3. The Arrow Technique (Shortcut for $2 \times 2$ and $3 \times 3$)

There is a visual shortcut for evaluating determinants, but it only works for $2 \times 2$ and $3 \times 3$ matrices! Do not attempt this on a $4 \times 4$.

1. Copy the first two columns of the matrix and paste them to the immediate right of the matrix.
2. Draw three diagonal arrows pointing down and to the right. Multiply the numbers along each arrow, and add these products together.
3. Draw three diagonal arrows pointing up and to the right. Multiply the numbers along each arrow, and subtract these products from your total.

4. Evaluating Determinants by Row Reduction

Sometimes cofactor expansion is too tedious. You can actually use elementary row operations to find the determinant, but you have to keep track of how the operations change the determinant value!

The Theorems

Let $A$ be a square matrix:

Multiply a Row: If you multiply a single row (or column) of $A$ by a constant $k$ to get a new matrix $B$, then $\det(B) = k \cdot \det(A)$.

Swap Rows: If you interchange two rows (or columns) to get $B$, then the sign flips: $\det(B) = -\det(A)$.

Add Rows (The Best One!): If you add a constant times one row to another row to get $B$, the determinant does not change at all: $\det(B) = \det(A)$.

The Scalar Theorem (Extremely Important)

If you multiply an entire $n \times n$ matrix $A$ by a scalar $k$, you are essentially multiplying every single row by $k$. Because of rule #1 above, the scalar $k$ gets pulled out $n$ times!

5. Triangular Matrices and Determinant Properties

Triangular Matrices

A matrix is considered triangular if all entries below the main diagonal are zero (upper triangular) or all entries above the main diagonal are zero (lower triangular).

Theorem: If $A$ is a triangular matrix, then $\det(A)$ is simply the product of the entries on its main diagonal!

Other Crucial Determinant Properties

Let $A$ and $B$ be square matrices of the same size:

Transpose: The determinant of a transpose is the same as the original matrix:

Product Rule: The determinant of a product is the product of the determinants:

Inverse Rule: If $A$ is invertible ($\det(A) \neq 0$), then:

Addition Warning: In general, $\det(A+B) \neq \det(A) + \det(B)$. Be careful not to distribute determinants over addition or subtraction!