Matrices and Determinants

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your ECAT exam.

A matrix is a rectangular array of numbers. A determinant is a scalar value computed from a square matrix that encodes important properties of the matrix.

Matrix Types:

• Square matrix: $n \times n$ (same rows and columns)
• Row matrix: $1 \times n$ (single row)
• Column matrix: $n \times 1$ (single column)
• Zero matrix: all elements are 0
• Identity matrix $I$: diagonal elements = 1, off-diagonal = 0
• Diagonal matrix: off-diagonal elements = 0
• Transpose $A^T$: rows become columns

Determinant of a $2 \times 2$ Matrix: $$A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \quad |A| = \det(A) = ad - bc$$

Determinant of a $3 \times 3$ Matrix (Sarrus’ Rule):

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

Or by expansion: $$\det(A) = a\begin{vmatrix} e & f \ h & i \end{vmatrix} - b\begin{vmatrix} d & f \ g & i \end{vmatrix} + c\begin{vmatrix} d & e \ g & h \end{vmatrix}$$

⚡ ECAT exam tips:

• If $|A| = 0$: the matrix is singular — no inverse exists
• If $|A| \neq 0$: the matrix is non-singular — inverse exists
• For a $2 \times 2$ matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$: inverse = $\frac{1}{ad-bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}$
• For triangular matrices (upper or lower), determinant = product of diagonal elements

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🟡 Standard — Regular Study (2d–2mo)

For ECAT students who want genuine understanding.

Properties of Determinants:

1. $\det(AB) = \det(A) \cdot \det(B)$
2. $\det(A^T) = \det(A)$
3. $\det(kA) = k^n \det(A)$ for $n \times n$ matrix
4. Swapping two rows changes sign of determinant
5. Adding a multiple of one row to another leaves determinant unchanged
6. If a row is all zeros: $\det = 0$
7. If two rows are identical: $\det = 0$

Matrix Operations:

Addition: $A + B$ — element-wise (matrices must have same dimensions) Multiplication: $(AB){ij} = \sum_k A{ik} B_{kj}$

• Note: $AB \neq BA$ in general
• $AI = IA = A$ for identity matrix Inverse: $A^{-1} = \frac{1}{|A|} \text{adj}(A)$ where adj$(A)$ is the adjugate (transpose of cofactor matrix)

Cofactor Expansion:

For a $3 \times 3$ matrix $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix}$:

Cofactor $C_{ij} = (-1)^{i+j} M_{ij}$ where $M_{ij}$ is the minor (determinant of matrix with row $i$ and column $j$ removed).

Example: Expand along first row: $\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$.

Solving Linear Equations — Cramer’s Rule:

For $Ax = b$ where $A$ is an $n \times n$ matrix: $$x_i = \frac{\det(A_i)}{\det(A)}$$ where $A_i$ is $A$ with column $i$ replaced by $b$.

Example: Solve $2x + y = 5$ and $3x - y = 1$. $D = \begin{vmatrix} 2 & 1 \ 3 & -1 \end{vmatrix} = -2 - 3 = -5$. $D_x = \begin{vmatrix} 5 & 1 \ 1 & -1 \end{vmatrix} = -5 - 1 = -6$. $D_y = \begin{vmatrix} 2 & 5 \ 3 & 1 \end{vmatrix} = 2 - 15 = -13$. $x = D_x/D = (-6)/(-5) = 6/5$. $y = D_y/D = (-13)/(-5) = 13/5$.

⚡ Common student mistakes:

1. Confusing matrix multiplication with element-wise multiplication — they are different
2. Forgetting that $AB \neq BA$ — matrix multiplication is not commutative
3. Not computing the sign $(-1)^{i+j}$ in cofactor expansion
4. Computing the adjugate wrong — it’s the transpose of the cofactor matrix

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for ECAT mastery of matrices and determinants.

Adjugate and Inverse:

The adjugate (or classical adjoint) of $A$: adj$(A) = C^T$ where $C_{ij}$ are cofactors. Then: $A^{-1} = \frac{1}{|A|} \text{adj}(A)$.

Also: $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| I$.

This gives a formula for the inverse without row operations.

Eigenvalues and Eigenvectors:

For $A\vec{v} = \lambda \vec{v}$ where $\vec{v} \neq \vec{0}$:

• $\lambda$ is an eigenvalue
• $\vec{v}$ is the corresponding eigenvector

The eigenvalues satisfy the characteristic equation: $$\det(A - \lambda I) = 0$$

For a $2 \times 2$ matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$: $\det\begin{pmatrix} a-\lambda & b \ c & d-\lambda \end{pmatrix} = (a-\lambda)(d-\lambda) - bc = \lambda^2 - (a+d)\lambda + (ad-bc) = 0$.

Sum of eigenvalues (trace): $\lambda_1 + \lambda_2 = a + d = \text{tr}(A)$. Product of eigenvalues: $\lambda_1 \lambda_2 = ad - bc = \det(A)$.

Matrix of Linear Transformation:

Every linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$ can be represented by a matrix $A$ such that $T(\vec{x}) = A\vec{x}$.

Rotation by angle $\theta$: $$R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix}$$

Reflection across x-axis: $$M_x = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}$$

Scaling by factor $k$: $$S_k = \begin{pmatrix} k & 0 \ 0 & k \end{pmatrix} = kI$$

Rank of a Matrix:

The rank $r(A)$ is the maximum number of linearly independent rows (or columns).

• $r(A) = n$ (full rank) if $|A| \neq 0$ for $n \times n$.
• $r(A) < n$ if $|A| = 0$.

Consistency of Linear Equations:

For $Ax = b$:

• If $r(A) = r([A|b])$: system is consistent. If $r(A) = n$: unique solution. If $r(A) < n$: infinitely many solutions.
• If $r(A) \neq r([A|b])$: system is inconsistent (no solution).

Inverse Using Row Operations (Gauss-Jordan):

$[A|I] \xrightarrow{\text{row ops}} [I|A^{-1}]$

This is often faster than the adjugate formula for large matrices.

Cayley-Hamilton Theorem:

Every square matrix satisfies its own characteristic equation: If $\det(A - \lambda I) = \lambda^n + c_{n-1}\lambda^{n-1} + … + c_1\lambda + c_0 = 0$, then: $$A^n + c_{n-1}A^{n-1} + … + c_1A + c_0 I = O$$

For $2 \times 2$: $A^2 - \text{tr}(A)A + \det(A)I = O$.

ECAT Previous Year Patterns:

• Determinant calculation: very common
• Matrix multiplication: common
• Inverse of matrix: common
• Solving linear equations: common
• Properties of determinants: periodic

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