Unit 6: Matrices and determinants

MTH110 — Remedial Mathematics 7 min read

A matrix is a rectangular array of numbers arranged in horizontal rows and vertical columns, enclosed in brackets and treated as a single algebraic object. The concept underlies the compact handling of linear systems, transformations and data tables; everything in this unit depends on the ideas of order, elements and the square-matrix condition.

Notation: A matrix is written $A = [a_{ij}]$ , where $a_{ij}$ is the element in row $i$ , column $j$ ; e.g. $a_{23}$ sits in row 2, column 3.
Order (size): An arrangement of $m$ rows and $n$ columns has order $m \times n$ , read "m by n". A $2\times3$ matrix has 6 elements.
Square matrix: Order $n \times n$ ; only square matrices have determinants and inverses.
Equality convention: Two matrices are equal only if they have the same order and $a_{ij}=b_{ij}$ for every position.
Principal diagonal: In a square matrix, the elements $a_{11}, a_{22}, \dots, a_{nn}$ .

II. Types of Matrices

Matrices are classified by shape and by the pattern of their elements, and the classification decides which operations apply.

A. types of matrices

Row matrix: A single row, order $1 \times n$ ; e.g. $[\,2\ 5\ 7\,]$ .
Column matrix: A single column, order $m \times 1$ ; e.g. $\begin{bmatrix}3\\ -1\end{bmatrix}$ .
Square matrix: Rows equal columns ( $n \times n$ ), the precondition for determinant and inverse.
Diagonal matrix: Square, with every off-diagonal element zero, e.g. $\begin{bmatrix}4&0\\0&9\end{bmatrix}$ .
Scalar matrix: Diagonal matrix with equal diagonal entries, e.g. $\begin{bmatrix}5&0\\0&5\end{bmatrix}$ .
Identity (unit) matrix $I$ : Scalar matrix with diagonal entries all 1; acts as the multiplicative identity, $AI = IA = A$ .
Zero (null) matrix $O$ : Every element is 0; the additive identity.
Upper/lower triangular: All elements below (or above) the diagonal are zero.
Symmetric matrix: $A^{T}=A$ , so $a_{ij}=a_{ji}$ .
Skew-symmetric matrix: $A^{T}=-A$ , forcing the diagonal to be zero.

III. Matrix Operations

Operations combine matrices under strict order rules; addition and subtraction need identical order, while multiplication needs a matching inner dimension.

A. addition of matrices

Addition is defined element by element for matrices of the same order.

Rule: $(A+B)_{ij} = a_{ij}+b_{ij}$ , valid only when $A$ and $B$ share the same order.
Properties: Commutative, $A+B=B+A$ ; associative, $(A+B)+C = A+(B+C)$ ; identity $A+O=A$ .
Example:

TEXT

[1 2]   [5 6]   [6  8]
[3 4] + [7 8] = [10 12]

B. subtraction of matrices

Subtraction is addition of the negative, again element by element on matrices of equal order.

Rule: $(A-B)_{ij} = a_{ij}-b_{ij}$ .
Not commutative: $A-B \neq B-A$ in general.
Example:

TEXT

[5 6]   [1 2]   [4 4]
[7 8] - [3 4] = [4 4]

C. multiplication of matrices

Two matrices multiply when the number of columns of the first equals the number of rows of the second; each product entry is a row-by-column dot product.

Conformability: $A_{m\times n}\, B_{n\times p} = C_{m\times p}$ ; the inner dimensions ( $n$ ) must match.
Rule: $c_{ij} = \sum_{k=1}^{n} a_{ik}\,b_{kj}$ , i.e. sum of products of row $i$ of $A$ with column $j$ of $B$ .
Scalar multiplication: $kA$ multiplies every element by $k$ .
Properties: Associative and distributive, but not commutative: $AB \neq BA$ usually.
Worked example:

TEXT

[1 2] [5 6]   [1·5+2·7  1·6+2·8]   [19 22]
[3 4] [7 8] = [3·5+4·7  3·6+4·8] = [43 50]

D. transpose of matrices

The transpose interchanges rows and columns, turning an $m\times n$ matrix into $n\times m$ .

Rule: $A^{T}=[a_{ji}]$ ; the element in row $i$ , column $j$ moves to row $j$ , column $i$ .
Properties: $(A^{T})^{T}=A$ ; $(A+B)^{T}=A^{T}+B^{T}$ ; the reversal law $(AB)^{T}=B^{T}A^{T}$ .
Example: $\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}^{T}=\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}$ .

IV. Determinants of Second and Third Order

A determinant is a single scalar computed from a square matrix, written $\det A$ or $|A|$ ; it measures whether the matrix is invertible ( $|A|\neq 0$ ) and scales the associated linear transformation.

A. determinant of second order

For a $2\times2$ matrix the determinant is the difference of diagonal products.

Formula: for $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ ,

TEXT

|A| = ad − bc

Symbols: $a,d$ lie on the principal diagonal; $b,c$ on the secondary diagonal.
Example: $\begin{vmatrix}3&2\\1&4\end{vmatrix}=3\cdot4-2\cdot1=10$ .

B. determinant of third order

A $3\times3$ determinant is evaluated by expansion along a row or column using minors and cofactors.

Minor $M_{ij}$ : the $2\times2$ determinant left after deleting row $i$ and column $j$ .
Cofactor: $C_{ij}=(-1)^{i+j}M_{ij}$ ; the sign pattern is $\begin{bmatrix}+&-&+\\-&+&-\\+&-&+\end{bmatrix}$ .
Expansion along row 1: for $A=\begin{bmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{bmatrix}$ ,

TEXT

|A| = a1(b2c3 − b3c2) − b1(a2c3 − a3c2) + c1(a2b3 − a3b2)

Example: $\begin{vmatrix}1&2&3\\0&4&5\\1&0&6\end{vmatrix}=1(24-0)-2(0-5)+3(0-4)=24+10-12=22$ .

C. properties and significance

Row/column swap: interchanging two rows changes the sign of $|A|$ .
Zero determinant: signals a singular (non-invertible) matrix and linearly dependent rows.
Transpose: $|A^{T}|=|A|$ .

V. Adjoint and Inverse of Non-Singular Matrices

A non-singular matrix is a square matrix with $|A|\neq 0$ ; only such matrices possess an inverse $A^{-1}$ satisfying $AA^{-1}=A^{-1}A=I$ , and the inverse is built from the adjoint.

A. adjoint of second and third order

The adjoint is the transpose of the cofactor matrix.

Definition: $\text{adj}\,A = [C_{ij}]^{T}$ , where $C_{ij}$ are the cofactors.
Second order shortcut: for $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ , swap the diagonal entries and negate the off-diagonal:

TEXT

adj A = [ d  −b ]
        [−c   a ]

Third order: compute all nine cofactors $C_{ij}=(-1)^{i+j}M_{ij}$ , arrange as a matrix, then transpose.
Key relation: $A\,(\text{adj}\,A) = |A|\,I$ .

B. inverse of non-singular matrices of second and third order

The inverse divides the adjoint by the determinant.

Formula:

TEXT

A⁻¹ = (1 / |A|) · adj A ,  provided |A| ≠ 0

Symbols: $|A|$ is the determinant, $\text{adj}\,A$ the adjoint; if $|A|=0$ the inverse does not exist.
Second-order example: for $A=\begin{bmatrix}3&2\\1&4\end{bmatrix}$ , $|A|=10$ , $\text{adj}\,A=\begin{bmatrix}4&-2\\-1&3\end{bmatrix}$ , so

TEXT

A⁻¹ = (1/10) [ 4  −2 ]  =  [ 0.4  −0.2]
             [−1   3 ]     [−0.1   0.3]

Property: $(AB)^{-1}=B^{-1}A^{-1}$ , and $(A^{-1})^{-1}=A$ .

VI. Solution of System of Linear Equations Using Inverse of a Matrix

A linear system can be packed into a matrix equation and solved in one step once the coefficient matrix is inverted.

A. solution of system of linear equations using inverse of a matrix

The method rewrites the system as $AX=B$ and isolates $X$ .

Matrix form: $AX=B$ , where $A$ is the coefficient matrix, $X$ the column of unknowns, $B$ the column of constants.
Solution: premultiply by $A^{-1}$ : $X = A^{-1}B$ , valid only when $|A|\neq 0$ (unique solution).
Singular case: if $|A|=0$ the system has either no solution or infinitely many.
Worked example: solve $2x+y=5,\ x+3y=10$ .

TEXT

A = [2 1]  B = [ 5]   |A| = 6−1 = 5
    [1 3]      [10]
adj A = [ 3 −1]   A⁻¹ = (1/5)[ 3 −1]
        [−1  2]              [−1  2]
X = A⁻¹B = (1/5)[3·5 −1·10] = (1/5)[ 5] = [1]
                [−1·5 +2·10]        [15]   [3]

So $x=1,\ y=3$ .

VII. Application of Determinant: Cramer's Rule

Cramer's rule solves a linear system directly from determinants, replacing one column of the coefficient matrix at a time with the constant column.

A. application of determinant to solve simultaneous equations by cramer's rule

Each unknown equals a ratio of determinants, so no matrix inversion is required.

Two variables: for $a_1x+b_1y=c_1,\ a_2x+b_2y=c_2$ ,

TEXT

D  = |a1 b1|   Dx = |c1 b1|   Dy = |a1 c1|
     |a2 b2|        |c2 b2|        |a2 c2|
x = Dx / D ,  y = Dy / D ,  D ≠ 0

Three variables: $D$ is the $3\times3$ coefficient determinant; $D_x, D_y, D_z$ replace the first, second and third columns respectively with the constants, giving $x=D_x/D$ , $y=D_y/D$ , $z=D_z/D$ .
Consistency test: $D\neq 0$ gives a unique solution; $D=0$ with some $D_x,D_y,D_z\neq 0$ means no solution; all zero means infinitely many.
Worked example: solve $2x+y=5,\ x+3y=10$ .

TEXT

D  = |2 1| = 5   Dx = |5  1| = 5   Dy = |2  5| = 15
     |1 3|            |10 3|            |1 10|
x = 5/5 = 1 ,  y = 15/5 = 3

Comparison with the inverse method:
1. Inverse method: finds all unknowns at once through $A^{-1}B$ ; efficient when the same $A$ recurs with different $B$ .
2. Cramer's rule: computes each unknown from its own determinant; cleaner for small systems but costly as $n$ grows because each variable needs a full determinant.

Unit 5

Unit 6: Matrices and determinants

Table of Contents

II. Types of Matrices

A. types of matrices

III. Matrix Operations

A. addition of matrices

B. subtraction of matrices

C. multiplication of matrices

D. transpose of matrices

IV. Determinants of Second and Third Order

A. determinant of second order

B. determinant of third order

C. properties and significance

V. Adjoint and Inverse of Non-Singular Matrices

A. adjoint of second and third order

B. inverse of non-singular matrices of second and third order

VI. Solution of System of Linear Equations Using Inverse of a Matrix

A. solution of system of linear equations using inverse of a matrix

VII. Application of Determinant: Cramer's Rule

A. application of determinant to solve simultaneous equations by cramer's rule

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