2.3 Matrices, Determinants, Inverses & Systems of Linear Equations

Matrices, Determinants, Inverses & Systems of Linear Equations

Matrices and determinants are a core FSc Part I Mathematics topic, and UET's admission FAQ names linear algebra among the ECAT Mathematics topics. Questions frequently demand fast classification of matrix symmetries, evaluation of determinants via row and column operations, rapid calculation of matrix inverses, and rigorous classification of solutions for homogeneous and non-homogeneous linear systems.

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1. Classification of Square Matrices

Let $A = [a_{ij}]$ be a square matrix of order $n \times n$ with real or complex entries.

Real Matrix Classifications

1. Symmetric Matrix: A real matrix $A$ is symmetric if its transpose equals itself:

   $A^T = A \iff a_{ij} = a_{ji} \quad \forall i, j$

2. Skew-Symmetric Matrix: A matrix $A$ is skew-symmetric if:

   $A^T = -A \iff a_{ij} = -a_{ji} \quad \forall i, j$

   Setting $i = j$ along the principal diagonal gives $a_{ii} = -a_{ii} \implies 2a_{ii} = 0 \implies a_{ii} = 0$. Thus, all diagonal elements of a skew-symmetric matrix must be zero.

   Theorem: Every square matrix $A$ can be uniquely decomposed into the sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$:

   $A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$

3. Orthogonal Matrix: A square matrix $A$ is orthogonal if its transpose equals its inverse:

   $A A^T = A^T A = I \implies A^{-1} = A^T$

   Taking the determinant of both sides: $|A A^T| = |A| |A^T| = |A|^2 = |I| = 1 \implies |A| = \pm 1$.

Complex Matrix Classifications

Let $A^\dagger = (\overline{A})^T$ denote the conjugate transpose (Hermitian conjugate) of $A$:

1. Hermitian Matrix: $A^\dagger = A \iff a_{ij} = \overline{a_{ji}}$ For diagonal elements: $a_{ii} = \overline{a_{ii}}$, which proves that all diagonal elements of a Hermitian matrix must be purely real.

2. Skew-Hermitian Matrix: $A^\dagger = -A \iff a_{ij} = -\overline{a_{ji}}$ For diagonal elements: $a_{ii} = -\overline{a_{ii}}$, proving that all diagonal elements of a skew-Hermitian matrix must be either 0 or purely imaginary.

3. Unitary Matrix: $A^\dagger A = A A^\dagger = I \implies |A| = e^{i\phi} \implies \lvert |A| \rvert = 1$

Special Algebraic Forms

• Idempotent Matrix: $A^2 = A$.
• Involutory Matrix: $A^2 = I \implies A^{-1} = A$.
• Nilpotent Matrix: $A^k = 0$ for some positive integer $k$; the smallest such $k$ is the index of nilpotency. (Every eigenvalue and the determinant of a nilpotent matrix are 0).

Matrix Symmetries Summary

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2. Properties of Determinants

The determinant $|A| = \det(A)$ is a scalar function mapping an $n \times n$ matrix to a number. Understanding its algebraic properties is key to solving ECAT questions without expanding cumbersome polynomials.

Core Properties of Determinants

1. Scalar Multiplication: For an $n \times n$ matrix $A$ and scalar $k$:

   $|kA| = k^n |A|$

   (A common student mistake is writing $|kA| = k|A|$; remember that $k$ is factored out of all $n$ rows simultaneously).

2. Multiplicative Identity: For square matrices $A$ and $B$ of the same order:

   $|AB| = |A| |B|$

3. Transposition Invariance: The determinant is invariant under transposition:

   $|A^T| = |A|$

4. Row/Column Interchange: Swapping any two rows or any two columns changes the algebraic sign of the determinant:

   $R_i \leftrightarrow R_j \implies |A'| = -|A|$

5. Vanishing Determinant Conditions: $|A| = 0$ if:

   • An entire row or column consists solely of zeros.
   • Any two rows or two columns are identical ($R_i = R_j$).
   • Any two rows or two columns are proportional ($R_i = k R_j$).

6. Elementary Row Operations: Adding a scalar multiple of one row to another row leaves the determinant completely unchanged:

   $R_i \to R_i + k R_j \implies |A'| = |A|$

7. Triangular Matrices: For an upper triangular, lower triangular, or diagonal matrix, the determinant is the product of its main diagonal elements:

   $|A| = a_{11} \cdot a_{22} \dots a_{nn} = \prod_{i=1}^n a_{ii}$

8. Inverse Determinant: For a non-singular matrix $A$:

   $|A^{-1}| = \frac{1}{|A|}$

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3. Minors, Cofactors, Adjoint, and Inverses

Minors and Cofactors

• The minor $M_{ij}$ of an entry $a_{ij}$ is the determinant of the $(n-1) \times (n-1)$ submatrix left after striking out row $i$ and column $j$.
• The cofactor $C_{ij}$ is defined with its signed parity:

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

Expansion Theorem

Expanding along row $i$:

$|A| = \sum_{j=1}^n a_{ij} C_{ij} = a_{i1} C_{i1} + a_{i2} C_{i2} + \dots + a_{in} C_{in}$

Orthogonality Property of Cofactors: If the elements of a row are multiplied by the cofactors of a different row, the sum is identically zero:

$\sum_{j=1}^n a_{ij} C_{kj} = 0 \quad \text{for } i \neq k$

The Adjoint Matrix

The adjoint (or adjugate) $\text{adj}(A)$ is the transpose of the cofactor matrix:

$\text{adj}(A) = [C_{ij}]^T = \begin{pmatrix} C_{11} & C_{21} & \dots & C_{n1} \\ C_{12} & C_{22} & \dots & C_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ C_{1n} & C_{2n} & \dots & C_{nn} \end{pmatrix}$

Fundamental Adjoint Identities

1. $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| I_n$
2. Determinant of the adjoint:

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

1. Adjoint of an adjoint:

$\text{adj}(\text{adj}(A)) = |A|^{n-2} A$

$|\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2}$

Matrix Inversion

A square matrix $A$ is invertible (non-singular) if and only if $|A| \neq 0$. The inverse $A^{-1}$ satisfies $A A^{-1} = A^{-1} A = I$ and is given by:

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

For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $|A| = ad - bc \neq 0$, the inverse shortcut is:

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

(Swap main diagonal elements, negate off-diagonal elements, divide by determinant).

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4. Systems of Linear Equations

A system of $m$ linear equations in $n$ unknowns can be represented in matrix form:

$AX = B$

where $A$ is the $m \times n$ coefficient matrix, $X = [x_1, x_2, \dots, x_n]^T$ is the column vector of unknowns, and $B = [b_1, b_2, \dots, b_m]^T$ is the constant vector.

1. Matrix Inversion Method ($n \times n$)

If $A$ is non-singular ($|A| \neq 0$):

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

2. Cramer's Rule

If $|A| \neq 0$, the unique solution is given by:

$x_i = \frac{|A_i|}{|A|}, \quad i = 1, 2, \dots, n$

where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with the constant vector $B$.

3. Consistency and Solvability Criteria

For an $n \times n$ non-homogeneous system $AX = B$:

1. Unique Solution (Consistent & Independent): $|A| \neq 0$ The rank of the coefficient matrix equals the rank of the augmented matrix: $\text{rank}(A) = \text{rank}([A|B]) = n$.

2. Infinitely Many Solutions (Consistent & Dependent): $|A| = 0 \quad \text{and} \quad \text{rank}(A) = \text{rank}([A|B]) < n$ The determinant check $\text{adj}(A) \cdot B = 0$ (equivalently $|A_1| = |A_2| = \dots = |A_n| = 0$) is necessary but not sufficient for $3 \times 3$ systems. For example, $x + y + z = 1$, $x + y + z = 2$, $x + y + z = 3$ gives $|A| = 0$, $\text{adj}(A) = 0$ and every $|A_i| = 0$, yet the three parallel planes have no common point. When the determinant test is inconclusive, row-reduce the augmented matrix and compare ranks.

3. No Solution (Inconsistent): $\text{rank}(A) < \text{rank}([A|B])$ A quick sufficient test: if $|A| = 0$ and $\text{adj}(A) \cdot B \neq 0$ (at least one $|A_i| \neq 0$), the system is certainly inconsistent.

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5. Homogeneous Linear Systems ($AX = 0$)

When $B = 0$, the system is termed homogeneous:

$a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = 0$

1. Trivial Solution: $x_1 = x_2 = \dots = x_n = 0$ is always a valid solution.
2. Non-Trivial Solutions: Non-zero solutions exist if and only if the coefficient matrix is singular:

$|A| = 0$

When $|A| = 0$, the system has an infinite number of non-trivial solutions, and the dimension of the solution space is the nullity $n - \text{rank}(A)$.

ECAT Recurring Pattern: Problems asking for a parameter $\lambda$ such that a homogeneous system has "non-zero solutions" or "non-trivial solutions" simply require setting $|A| = 0$ and solving the resulting polynomial for $\lambda$.

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6. High-Speed ECAT Determinant Shortcuts

Shortcut 1: Circulant Determinant

The cyclic determinant occurs frequently in ECAT algebra:

$\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = -(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = 3abc - a^3 - b^3 - c^3$

If $a + b + c = 0$, the determinant vanishes instantly.

Shortcut 2: Skew-Symmetric Matrices of Odd Order

If $A$ is skew-symmetric of order $n \times n$ where $n$ is odd, then:

$|A| = |A^T| = |-A| = (-1)^n |A| = -|A| \implies 2|A| = 0 \implies |A| = 0$

Any $3 \times 3$ or $5 \times 5$ skew-symmetric matrix has a determinant of 0 without calculating any products!

Shortcut 3: Linearly Dependent Rows/Columns via Inspection

Check if adding two rows creates a multiple of the third row. For instance, if $R_1 + R_2 = R_3$, the determinant is zero immediately.