Matrices and Determinants

Foundations: Structure and Operations

Matrices are rectangular arrays of numbers organized into rows and columns. Their order or dimension ($m \times n$) determines how they interact with other matrices through addition and scalar multiplication.

$$A + B = [a_{ij} + b_{ij}]$$

Matrix addition is performed element-wise by summing entries at the same index.

$a_{ij}$=Element in row $i$, column $j$(scalar)

$b_{ij}$=Corresponding element in matrix $B$(scalar)

Different Orders

→

Addition is undefined; matrices must be of the same dimension.

Element Indexing: $a_{ij}$ refers to the entry in the $i^{th}$ row and $j^{th}$ column.

Dimension Rule: Addition and subtraction require exactly matching $m \times n$ dimensions.

Commutativity: Matrix addition is commutative ($A + B = B + A$), but multiplication is NOT.

Additive Identity: The Null Matrix $O$ satisfies $A + O = A$ for any matrix of the same order.

Scalar Scaling: $k(A + B) = kA + kB$ and $(k + l)A = kA + lA$ — scalar multiplication distributes over both matrix and scalar addition.

Matrix Multiplication: The Row-by-Column Rule

Unlike addition, matrix multiplication is not element-wise. It involves the dot product of rows from the first matrix and columns from the second. The inner dimensions must match for the product to be conformable.

$$c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$

The result element $c_{ij}$ is the sum of products of row $i$ from $A$ and column $j$ from $B$.

$a_{ik}$=Elements of the $i^{th}$ row of matrix $A$(unitless)

$b_{kj}$=Elements of the $j^{th}$ column of matrix $B$(unitless)

$n$=Number of columns in $A$ (must equal rows in $B$)(integer)

$AB \neq BA$

→

Matrix multiplication is generally non-commutative.

Dimension Matching: To multiply $(m \times n)$ by $(n \times p)$, the inner $n$ values must be equal.

Resulting Order: The product matrix will have dimensions $(m \times p)$.

Identity Property: Multiplication by the Identity Matrix ($I$) leaves the original matrix unchanged.

Determinants and Singularity

The determinant is a scalar value that captures essential properties of a square matrix. It determines if a matrix is invertible; a matrix with a determinant of zero is called singular and has no inverse.

$$|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$$

Formula for a 3x3 matrix determinant via expansion by the first row.

$|A|$=Determinant of matrix $A$(scalar)

$(ei-fh)$=Minor of element $a$(scalar)

$|A| = 0$

→

The matrix is singular and cannot be inverted.

2×2 Formula: For $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is simply $ad - bc$ (main diagonal minus secondary diagonal).

Cofactor Sign Pattern: Expansion uses the checkerboard $(-1)^{i+j}$, starting with $+$ at position $(1,1)$.

Product Rule: The determinant of a product equals the product of the determinants: $|AB| = |A||B|$.

Expansion Choice: You may expand along any row or column — the result is always the same. Choose the row/column with the most zeros to save work.

Properties of Determinants

Several key properties of determinants allow evaluation without full expansion. These shortcuts reduce computation by identifying special structures in the matrix.

Transpose Property: $|A| = |A^T|$. The determinant is unchanged when rows and columns are swapped.

Row/Column Interchange: Swapping two rows (or columns) reverses the sign of the determinant.

Identical Rows/Columns: If any two rows (or columns) are identical, $|A| = 0$.

Zero Row/Column: If an entire row or column is zero, $|A| = 0$.

Scalar Factor: A common factor $k$ from any single row can be taken outside: $|kR_i| = k|A|$. For the entire matrix: $|kA| = k^n|A|$.

Row Addition: Adding a scalar multiple of one row to another does not change the determinant.

Triangular Matrix: The determinant of a triangular (or diagonal) matrix equals the product of its diagonal entries.

Special Types of Matrices

Matrices are classified by the relationships between their entries. A Symmetric Matrix satisfies $A^T = A$, while a Skew-Symmetric Matrix satisfies $A^T = -A$. Similarly, a Hermitian Matrix satisfies $A^\theta = A$ for complex entries.

Symmetric: $a_{ij} = a_{ji}$ for all entries. All diagonal matrices are symmetric.

Skew-Symmetric: $a_{ij} = -a_{ji}$, which forces all diagonal entries to be zero.

Hermitian: Like symmetric, but for complex matrices — $\bar{a}_{ji} = a_{ij}$.

Skew-Hermitian: $A^\theta = -A$; diagonal entries are either zero or purely imaginary.

Null Matrix: Every entry is zero. Also called the zero matrix $O$.

Scalar Matrix: A diagonal matrix where all diagonal entries are the same scalar $k$. Equivalent to $kI$.

The Adjoint and Inverse Matrix

The inverse of a matrix $A$ is another matrix $A^{-1}$ such that their product is the Identity Matrix. Finding the inverse requires the Adjoint matrix, which is the transpose of the cofactor matrix.

$$A^{-1} = \frac{1}{|A|} \text{adj}(A)$$

The inverse is the adjoint matrix scaled by the reciprocal of the determinant.

$A^{-1}$=Multiplicative Inverse Matrix(matrix)

$\text{adj}(A)$=Adjoint matrix (Transpose of Cofactors)(matrix)

$|A|$=Determinant of A (must be non-zero)(scalar)

$|A| = 0$

→

The matrix is singular and no inverse exists.

Existence: $A^{-1}$ exists if and only if $|A| \neq 0$.

Inverse Determinant: $|A^{-1}| = \frac{1}{|A|}$, since $|AA^{-1}| = |I| = 1$.

2×2 Adjoint Shortcut: For $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the adjoint is $\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ — swap diagonals, negate off-diagonals.

Verification: Always check your answer: $AA^{-1}$ must equal $I$.