2.6 Matrices and Determinants

Detailed Theory: Matrices and Determinants

1. Introduction to Matrices

1.1 What is a Matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.

Example:

$$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$

This is a $2 \times 3$ matrix (2 rows, 3 columns).

1.2 Notation

• Matrices are usually denoted by capital letters: $A, B, C, \ldots$

• The element in the $i$-th row and $j$-th column is denoted by $a_{ij}$

• A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix

General form:

$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$

1.3 Types of Matrices

a) Row Matrix

A matrix with only one row.

Example:

$$\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$$

($1 \times 3$)

b) Column Matrix

A matrix with only one column.

Example:

$$\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$$

($3 \times 1$)

c) Zero Matrix (Null Matrix)

A matrix with all elements zero. Denoted by $O$.

Example:

$$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$

($2 \times 2$ zero matrix)

d) Square Matrix

A matrix with same number of rows and columns ($m = n$).

Example:

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

($2 \times 2$)

e) Diagonal Matrix

A square matrix where all non-diagonal elements are zero.

Example:

$$\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$$

f) Scalar Matrix

A diagonal matrix where all diagonal elements are equal.

Example:

$$\begin{bmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{bmatrix}$$

g) Identity Matrix

A scalar matrix with diagonal elements = 1. Denoted by $I$ or $I_n$.

Examples:

$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

$$I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

h) Upper Triangular Matrix

A square matrix where all elements below the main diagonal are zero.

Example:

$$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}$$

i) Lower Triangular Matrix

A square matrix where all elements above the main diagonal are zero.

Example:

$$\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6 \end{bmatrix}$$

j) Symmetric Matrix

A square matrix that equals its transpose: $A^T = A$

Example:

$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$$

k) Skew-Symmetric Matrix

A square matrix that equals the negative of its transpose: $A^T = -A$ Diagonal elements must be zero.

Example:

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

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2. Matrix Operations

2.1 Equality of Matrices

Two matrices $A$ and $B$ are equal if:

1. They have same dimensions

2. Corresponding elements are equal

$$A = B \iff a_{ij} = b_{ij} \quad \text{for all } i, j$$

2.2 Addition of Matrices

Matrices of same dimensions can be added element-wise.

If $A = [a_{ij}]$ and $B = [b_{ij}]$ are both $m \times n$, then:

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

Properties:

1. Commutative: $A + B = B + A$

2. Associative: $(A + B) + C = A + (B + C)$

3. Additive Identity: $A + O = A$

4. Additive Inverse: $A + (-A) = O$

2.3 Subtraction of Matrices

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

2.4 Scalar Multiplication

If $k$ is a scalar and $A = [a_{ij}]$, then:

$$kA = [k \cdot a_{ij}]$$

Properties:

1. $k(A + B) = kA + kB$

2. $(k + l)A = kA + lA$

3. $k(lA) = (kl)A$

4. $1 \cdot A = A$

2.5 Matrix Multiplication

a) Condition for Multiplication

Matrix $A$ ($m \times n$) can multiply matrix $B$ ($p \times q$) only if:

$$n = p$$

The product $AB$ will have dimensions $m \times q$.

b) Multiplication Process

If $A = [a_{ij}]$ is $m \times n$ and $B = [b_{jk}]$ is $n \times p$, then:

$$C = AB \quad \text{where} \quad c_{ik} = \sum_{j=1}^{n} a_{ij} b_{jk}$$

Element $c_{ik}$ is dot product of $i$-th row of $A$ and $k$-th column of $B$.

Example:

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$$

$$AB = \begin{bmatrix} (1\times5 + 2\times7) & (1\times6 + 2\times8) \\ (3\times5 + 4\times7) & (3\times6 + 4\times8) \end{bmatrix}$$

$$AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$$

c) Properties of Matrix Multiplication

1. Not commutative: $AB \neq BA$ in general

2. Associative: $A(BC) = (AB)C$

3. Distributive: $A(B + C) = AB + AC$

4. Multiplicative Identity: $AI = IA = A$

5. Multiplication with zero matrix: $AO = OA = O$

2.6 Transpose of a Matrix

The transpose of matrix $A$, denoted $A^T$ or $A'$, is obtained by interchanging rows and columns.

If $A = [a_{ij}]$ is $m \times n$, then $A^T = [a_{ji}]$ is $n \times m$.

Properties:

1. $(A^T)^T = A$

2. $(A + B)^T = A^T + B^T$

3. $(kA)^T = kA^T$

4. $(AB)^T = B^T A^T$

2.7 Trace of a Matrix

For a square matrix $A$, the trace is sum of diagonal elements:

$$\text{tr}(A) = \sum_{i=1}^{n} a_{ii}$$

Properties:

1. $\text{tr}(A + B) = \text{tr}(A) + \text{tr}(B)$

2. $\text{tr}(kA) = k \cdot \text{tr}(A)$

3. $\text{tr}(AB) = \text{tr}(BA)$

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3. Determinants

3.1 Definition

Determinant is a scalar value computed from a square matrix.

Notation: $\det(A)$ or $|A|$

3.2 Determinant of 2×2 Matrix

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

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

3.3 Determinant of 3×3 Matrix

For $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$:

$$\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$$

This can be remembered as Sarrus' Rule:

Write first two columns again to the right:

$$\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$$

Sum of products of diagonals from left to right minus sum of products of diagonals from right to left.

3.4 Minors and Cofactors

a) Minor

The minor $M_{ij}$ of element $a_{ij}$ is the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column.

b) Cofactor

The cofactor $C_{ij}$ of element $a_{ij}$ is:

$$C_{ij} = (-1)^{i+j} M_{ij}$$

Example: For $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$:

Minor $M_{11}$ = determinant of $\begin{bmatrix} 5 & 6 \\ 8 & 9 \end{bmatrix} = 5\times9 - 6\times8 = 45 - 48 = -3$

Cofactor $C_{11} = (-1)^{1+1} M_{11} = (-1)^2 \times (-3) = -3$

3.5 Expansion by Cofactors

Determinant can be computed by expanding along any row or column:

Along $i$-th row:

$$\det(A) = \sum_{j=1}^{n} a_{ij} C_{ij}$$

Along $j$-th column:

$$\det(A) = \sum_{i=1}^{n} a_{ij} C_{ij}$$

3.6 Properties of Determinants

a) Basic Properties

1. $\det(I) = 1$

2. $\det(O) = 0$

3. $\det(A^T) = \det(A)$

4. If two rows (or columns) are identical, $\det(A) = 0$

5. If a row (or column) has all zeros, $\det(A) = 0$

b) Row/Column Operations

Let $A$ be an $n \times n$ matrix.

1. Row interchange: Swapping two rows changes sign of determinant

2. Scalar multiplication: Multiplying a row by $k$ multiplies determinant by $k$

3. Row addition: Adding a multiple of one row to another doesn't change determinant

c) Multiplication Property

$$\det(AB) = \det(A) \cdot \det(B)$$

d) Inverse Property

If $A$ is invertible,

$$\det(A^{-1}) = \frac{1}{\det(A)}$$

3.7 Special Determinants

a) Diagonal Matrix

Determinant = product of diagonal elements.

$$\det\begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} = abc$$

b) Triangular Matrix

Determinant = product of diagonal elements.

c) Vandermonde Determinant

$$\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{vmatrix} = (y-x)(z-x)(z-y)$$

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4. Inverse of a Matrix

4.1 Definition

A square matrix $A$ is invertible if there exists matrix $B$ such that:

$$AB = BA = I$$

$B$ is called the inverse of $A$, denoted $A^{-1}$.

Note: Only square matrices can be invertible, but not all square matrices are invertible.

4.2 Condition for Invertibility

A square matrix $A$ is invertible if and only if:

$$\det(A) \neq 0$$

If $\det(A) = 0$, $A$ is called singular or non-invertible.

4.3 Finding Inverse

a) Formula for 2×2 Matrix

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

If $\det(A) = ad - bc \neq 0$, then:

$$A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

Example: $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$

$$\det(A) = 2\times4 - 3\times1 = 8 - 3 = 5 \neq 0$$

$$A^{-1} = \frac{1}{5} \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} \frac{4}{5} & -\frac{3}{5} \\ -\frac{1}{5} & \frac{2}{5} \end{bmatrix}$$

b) Using Adjoint (for n×n)

The adjoint of $A$, denoted $\text{adj}(A)$, is the transpose of the cofactor matrix.

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

Steps:

1. Find cofactor matrix $C$ where $C_{ij} = (-1)^{i+j} M_{ij}$

2. Transpose to get adjoint: $\text{adj}(A) = C^T$

3. Divide by $\det(A)$

c) Using Elementary Row Operations (Gauss-Jordan)

Augment $A$ with $I$: $[A | I]$

Apply row operations to transform $A$ to $I$

The right side becomes $A^{-1}$: $[I | A^{-1}]$

4.4 Properties of Inverse

1. $(A^{-1})^{-1} = A$

2. $(AB)^{-1} = B^{-1} A^{-1}$

3. $(A^T)^{-1} = (A^{-1})^T$

4. $\det(A^{-1}) = \frac{1}{\det(A)}$

5. $(kA)^{-1} = \frac{1}{k} A^{-1}$ for $k \neq 0$

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5. Systems of Linear Equations

5.1 Matrix Representation

A system of $m$ linear equations in $n$ variables:

$$a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1$$

$$a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2$$

$\vdots$

$$a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m$$

Can be written as: $AX = B$

where

$$A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}, \quad B = \begin{bmatrix} b_1 \\ \vdots \\ b_m \end{bmatrix}$$

5.2 Solution Methods

a) Using Inverse (for n×n systems)

If $A$ is square and invertible, solution is:

$$X = A^{-1}B$$

b) Using Cramer's Rule

For system $AX = B$ where $A$ is $n \times n$ and $\det(A) \neq 0$:

$$x_i = \frac{\det(A_i)}{\det(A)}$$

where $A_i$ is matrix obtained by replacing $i$-th column of $A$ with $B$.

Example: Solve:

$$2x + 3y = 8$$

$$x + 4y = 6$$

In matrix form:

$$\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 6 \end{bmatrix}$$

$$\det(A) = 2\times4 - 3\times1 = 8 - 3 = 5$$

$$x = \frac{\begin{vmatrix} 8 & 3 \\ 6 & 4 \end{vmatrix}}{\det(A)} = \frac{32 - 18}{5} = \frac{14}{5}$$

$$y = \frac{\begin{vmatrix} 2 & 8 \\ 1 & 6 \end{vmatrix}}{\det(A)} = \frac{12 - 8}{5} = \frac{4}{5}$$

5.3 Consistency of Systems

For $AX = B$:

1. Consistent: Has at least one solution

2. Inconsistent: Has no solution

Rank Method: Let $[A|B]$ be augmented matrix.

System is:

• Consistent if $\text{rank}(A) = \text{rank}([A|B])$

• Inconsistent if $\text{rank}(A) \neq \text{rank}([A|B])$

If consistent:

• Unique solution if $\text{rank}(A) = n$ (number of variables)

• Infinite solutions if $\text{rank}(A) < n$

5.4 Homogeneous Systems

System $AX = O$ (all $b_i = 0$)

Properties:

1. Always consistent (trivial solution $X = O$ exists)

2. Has non-trivial solutions if and only if $\det(A) = 0$

3. If non-trivial solutions exist, they are infinite in number

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6. Eigenvalues and Eigenvectors

6.1 Definition

For square matrix $A$, a non-zero vector $X$ is an eigenvector if:

$$AX = \lambda X \quad \text{for some scalar } \lambda$$

$\lambda$ is called the eigenvalue corresponding to eigenvector $X$.

6.2 Finding Eigenvalues

The equation $AX = \lambda X$ can be written as:

$$(A - \lambda I)X = O$$

For non-zero solution $X$, we need:

$$\det(A - \lambda I) = 0$$

This is called the characteristic equation.

6.3 Steps to Find Eigenvalues and Eigenvectors

1. Form $A - \lambda I$

2. Set $\det(A - \lambda I) = 0$, solve for $\lambda$ (eigenvalues)

3. For each $\lambda$, solve $(A - \lambda I)X = O$ to find eigenvectors

Example: Find eigenvalues and eigenvectors of $A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$

Step 1:

$$A - \lambda I = \begin{bmatrix} 2-\lambda & 1 \\ 1 & 2-\lambda \end{bmatrix}$$

Step 2:

$$\det(A - \lambda I) = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = 0$$

$$(\lambda-1)(\lambda-3) = 0$$

Eigenvalues: $\lambda_1 = 1$, $\lambda_2 = 3$

Step 3: For $\lambda_1 = 1$:

Solve

$$\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$$

Equation: $x + y = 0 \Rightarrow y = -x$

Eigenvector: $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$ or any multiple

For $\lambda_2 = 3$:

Solve

$$\begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$$

Equation: $-x + y = 0 \Rightarrow y = x$

Eigenvector: $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ or any multiple

6.4 Properties

1. Sum of eigenvalues = trace of $A$

2. Product of eigenvalues = determinant of $A$

3. Eigenvalues of diagonal/triangular matrix = diagonal elements

4. If $\lambda$ is eigenvalue of $A$, then for polynomial $p(A)$, $p(\lambda)$ is eigenvalue

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7. Rank of a Matrix

7.1 Definition

The rank of a matrix is the maximum number of linearly independent rows (or columns).

Notation: $\text{rank}(A)$

7.2 Finding Rank

a) Using Row Echelon Form

Transform matrix to row echelon form using elementary row operations.

Rank = number of non-zero rows.

b) Using Determinants

Rank is the size of largest non-zero minor.

7.3 Properties

1. $\text{rank}(A) = \text{rank}(A^T)$

2. $\text{rank}(A) \leq \min(m, n)$ for $m \times n$ matrix

3. If $\text{rank}(A) = n$ for $n \times n$ matrix, then $A$ is invertible

4. $\text{rank}(AB) \leq \min(\text{rank}(A), \text{rank}(B))$

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8. Special Matrices and Properties

8.1 Orthogonal Matrix

A square matrix $A$ is orthogonal if:

$$A^T A = AA^T = I$$

Equivalently: $A^T = A^{-1}$

Properties:

1. Columns (and rows) are orthonormal vectors

2. $\det(A) = \pm 1$

3. Preserves length: $\|AX\| = \|X\|$

8.2 Idempotent Matrix

A square matrix $A$ is idempotent if:

$$A^2 = A$$

Example:

$$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$

8.3 Nilpotent Matrix

A square matrix $A$ is nilpotent if:

$$A^k = O \quad \text{for some positive integer } k$$

Example:

$$\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$$

($A^2 = O$)

8.4 Involutory Matrix

A square matrix $A$ is involutory if:

$$A^2 = I$$

Equivalently: $A^{-1} = A$

Example:

$$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

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9. Matrix Equations

9.1 Solving $AX = B$

If $A$ is invertible: $X = A^{-1}B$

9.2 Solving $XA = B$

If $A$ is invertible: $X = BA^{-1}$

9.3 Sylvester Equation

$AX + XB = C$

Solution involves Kronecker product.

9.4 Lyapunov Equation

$A^TX + XA = -Q$

Important in control theory.

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10. Applications

10.1 Computer Graphics

Matrices used for transformations:

• Translation

• Rotation

• Scaling

• Shearing

10.2 Cryptography

Matrices used in encryption algorithms.

10.3 Economics

Input-output analysis using Leontief models.

10.4 Physics

Quantum mechanics, rotation matrices, stress tensors.

10.5 Statistics

Covariance matrices, multivariate analysis.

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11. Solved Examples

Example 1: Matrix Multiplication

If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, find $AB$ and $BA$.

Solution:

$$AB = \begin{bmatrix} 1\times5 + 2\times7 & 1\times6 + 2\times8 \\ 3\times5 + 4\times7 & 3\times6 + 4\times8 \end{bmatrix}$$

$$AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$$

$$BA = \begin{bmatrix} 5\times1 + 6\times3 & 5\times2 + 6\times4 \\ 7\times1 + 8\times3 & 7\times2 + 8\times4 \end{bmatrix}$$

$$BA = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}$$

Note: $AB \neq BA$

Example 2: Determinant Calculation

Find

$$\det\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$

Solution:

Using Sarrus' rule:

Sum of left-to-right diagonals:

$$(1\times5\times9) + (2\times6\times7) + (3\times4\times8) = 45 + 84 + 96 = 225$$

Sum of right-to-left diagonals:

$$(3\times5\times7) + (1\times6\times8) + (2\times4\times9) = 105 + 48 + 72 = 225$$

Determinant = $225 - 225 = 0$

Example 3: Inverse Calculation

Find inverse of $A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$

Solution:

$$\det(A) = 2\times3 - 5\times1 = 6 - 5 = 1$$

$$A^{-1} = \frac{1}{1} \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix}$$

Example 4: System of Equations

Solve using matrices:

$$2x + 3y = 11$$

$$x + 2y = 6$$

Solution:

Matrix form:

$$\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 11 \\ 6 \end{bmatrix}$$

$$\det(A) = 4 - 3 = 1 \neq 0$$

$$A^{-1} = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 11 \\ 6 \end{bmatrix} = \begin{bmatrix} 22-18 \\ -11+12 \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \end{bmatrix}$$

So $x=4$, $y=1$

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12. Important Formulas Summary

12.1 Determinants

• 2×2:

$$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$

• 3×3: Use Sarrus' rule or cofactor expansion

12.2 Inverse

• 2×2:

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

• General: $A^{-1} = \frac{1}{\det(A)}\text{adj}(A)$

12.3 Eigenvalues

Solve: $\det(A - \lambda I) = 0$

12.4 Trace

$$\text{tr}(A) = \sum a_{ii}$$

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13. Exam Tips and Common Mistakes

13.1 Common Mistakes

1. Matrix multiplication: Not checking dimensions compatibility

2. Inverse: Forgetting to check $\det(A) \neq 0$ first

3. Determinant: Incorrect sign in cofactor expansion

4. Eigenvectors: Forgetting eigenvectors are defined up to scalar multiple

5. Transpose: $(AB)^T = B^TA^T$ not $A^TB^T$

13.2 Problem-Solving Strategy

1. Identify matrix type and dimensions

2. Choose appropriate method (inverse, determinant, row operations)

3. Show all steps clearly

4. Check answer when possible (e.g., verify $AA^{-1} = I$)

13.3 Quick Checks

1. Square matrix needed for inverse, determinant, eigenvalues

2. $\det(A) \neq 0$ for invertibility

3. Dimensions must match for matrix operations

4. Eigenvectors are never zero vectors

This comprehensive theory covers all aspects of matrices and determinants with detailed explanations and examples, providing complete preparation for the entrance examination.