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02. Introduction to Matrices

Dated: 10-03-2025

Matrix

A matrix is a collection or an array of numbers1 or functions2 (both are called members, entry or elements of the matrix) arranged in rows and columns.

Denotation

The matrix is denoted by capital letters such as \(A, B\) etc meanwhile its members are represented by small letters such as \(a, b\) etc.

Order

The order is the size or dimension of a matrix, written as \(r \times c\) where \(r\) represents the number of rows and \(c\) represents the number of columns.

\[ A = \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{matrix} \right] \]

In general terms, the element in \(i^{\text{th}}\) row and \(j^\text{th}\) column of a \(m \times n\) matrix can be written as \(a_{\text{ij}}\).

Example

The following matrix has an order of \(2 \times 3\) with \(2\) rows and \(3\) columns.

\[ A = \left[ \begin{matrix} 2 & -1 & 3\\ 0 & 4 & 6 \end{matrix} \right] \]

Square Matrix

A matrix is called a square matrix if the number of rows is equal to the number of columns.

\[ A = \left[ \begin{matrix} 1 & 2\\ 3 & 4 \end{matrix} \right] \]

Equal Matrices

Two matrices are called equal matrices if

  • They have same dimensions.
  • The corresponding entries are equal.
\[ A = \left[ \begin{matrix} 1 & 2\\ 3 & 4 \end{matrix} \right] \quad \text{ and } \quad B = \left[ \begin{matrix} 1 & 2\\ 3 & 4 \end{matrix} \right] \]

Both are equal matrices.

Column Matrix

A matrix is called a column matrix if it has only one column and \(n\) rows.

\[X = \left[ \begin{matrix} a_{11}\\ a_{21}\\ \vdots\\ a_{n1}\\ \end{matrix} \right] = [a_{\text{i1}}]_{n \times 1} \]

A column matrix is also called a column vector or simply a vector.3

Multiple of a Matrix

If \(A\) is a matrix and \(k\) is a non-zero constant then

\[ kA = k\left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{matrix} \right] \]
\[ = \left[ \begin{matrix} ka_{11} & ka_{12} & \cdots & ka_{1n} \\ ka_{21} & ka_{22} & \cdots & ka_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ka_{m1} & ka_{m2} & \cdots & ka_{mn} \end{matrix} \right] \]
\[= [ka_{\text{ij}}]_{m \times n}\]
Commutative property of scalar multiplication
\[kA = Ak\]

Addition of Matrices

If \(A = [a_{\text{ij}}]\) and \(B = [b_{\text{ij}}]\) are matrices of order \(m \times n\) then \(A + B = [a_{\text{ij}} + b_{\text{ij}}]_{m \times n}\).

Difference of Matrices

Similar to addition, \(A - B\) can be written as \(A + (-B)\) where \(-B = -1 \times B\).

Product of Matrices

\[A_{m \times n} \cdot B_{n \times p} = C_{m \times p}\]
\[ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \]
\[ B = \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{np} \end{bmatrix} \]
\[A \times B = AB = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{np} \end{bmatrix} \]
\[ = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} + \cdots + a_{1n}b_{n1} & \cdots & a_{11}b_{1p} + a_{12}b_{2p} + \cdots + a_{1n}b_{np}\\ a_{21}b_{11} + a_{22}b_{21} + \cdots + a_{2n}b_{n1} & \cdots & a_{21}b_{1p} + a_{22}b_{2p} + \cdots + a_{2n}b_{np}\\ \vdots & \vdots & \vdots \\ a_{m1}b_{11} + a_{m2}b_{21} + \cdots + a_{mn}b_{n1} & \cdots & a_{m1}b_{1p} + a_{m2}b_{2p} + \cdots + a_{mn}b_{np}\\ \end{bmatrix} \]
\[ = \begin{bmatrix} \sum_{k = 1}^n a_{ik}b_{kj} \end{bmatrix} _{m \times p} \]

\(AB \neq BA\)

For two matrices to be eligible for multiplication, the number of columns of the first one has to be equal to number of rows of second one.

\[A_{m \times n} \cdot B_{n \times p} = C_{m \times p}\]

Multiplicative Identity

\[ I = \left[ \begin{matrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{matrix} \right] \]
\[A \times I = I \times A = A\]

Zero or Null Matrix

A matrix with all entries being \(0\) is called a null matrix.

\[ O = \left[ \begin{matrix} 0 & 0\\ 0 & 0 \end{matrix} \right] \]
\[ O = \left[ \begin{matrix} 0\\ 0 \end{matrix} \right] \]

Associative Law

If \(A, B\) and \(C\) are three matrices then

\[A(BC) = (AB)C\]

Distributive Law

If \(A, B\) and \(C\) are three matrices then

\[A(B + C) = AB + AC\]
\[(A + B)C = AC + BC\]

Remarks

There are some rules from \(\mathbb R\) which do not carry over to matrix addition.
For example, \(\forall a, b, c\in \mathbb R\),

  1. \(ab = ac \land a \neq 0 \implies b = c\) (Law of Cancellation)
  2. \(ab = 0 \implies a = 0 \lor b = 0\).
Law of Cancellation only works when the cancelled matrix is invertible.4

Meaning, \(AB = AC \implies A^{-1}AB = A^{-1}AC \implies B = C \iff A^{-1} \text{ exists}\).
Cancellation law only partially works.

Determinant of a Matrix

The determinant is a special number associated with every square matrix and is denoted by \(\det(A)\) or \(|A|\).

\[ A_{2 \times 2} = \left[ \begin{matrix} 1 & 2\\ 3 & 4 \end{matrix} \right] \]
\[ |A_{2 \times 2}| = \left| \begin{matrix} 1 & 2\\ 3 & 4 \end{matrix} \right| = (1 \times 4) - (3 \times 2) = 4 - 6 = -2 \]
\[ A_{3 \times 3} = \left[ \begin{matrix} 3 & 6 & 2 \\ 2 & 5 & 1 \\ -1 & 2 & 4 \end{matrix} \right] \]
\[ |A_{3 \times 3}| = \left| \begin{matrix} 3 & 6 & 2 \\ 2 & 5 & 1 \\ -1 & 2 & 4 \end{matrix} \right| \]
\[ \begin{vmatrix} 3 & 6 & 2 \\ 2 & 5 & 1 \\ -1 & 2 & 4 \end{vmatrix} = 3\begin{vmatrix} 5 & 1 \\ 2 & 4 \end{vmatrix} - 6\begin{vmatrix} 2 & 1 \\ -1 & 4 \end{vmatrix} + 2\begin{vmatrix} 2 & 5 \\ -1 & 2 \end{vmatrix} \]
\[ 3(20 - 2) - 6(8 +1) + 2(4 + 5) = 18 \]

Transpose of a Matrix

If \(A_{m \times n}\) is a matrix with \(m\) rows and \(n\) columns then its transpose \(A_{m \times n}^T = A_{n \times m}\) can be obtained by interchanging its rows with columns.

\[ A = \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{matrix} \right] \]
\[ A^T = \left[ \begin{matrix} a_{11} & a_{21} & \cdots & a_{m1} \\ a_{12} & a_{22} & \cdots & a_{m2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{mn} \end{matrix} \right] \]

Properties

  • \((kA)^T= k A^T\)
  • \((A + B)^T = A^T + B^T\)
  • \((ABC)^T = C^TB^TA^T\)

Multiplicative Inverse

If \(A_{n \times n}\) and \(B_{n \times n}\) are matrices then \(B_{n \times n}\) is called the multiplicative inverse of \(A_{n \times n}\) if

\[AB = BA = I\]

And it is denoted as \(B = A^{-1}\)

Singular and Non Singular Matrix

A matrix \(A\) is called singular if \(\det(A) = 0\) otherwise if \(\det(A) \neq 0\) is called non singular matrix.

Minor of an Element of a Matrix

Let \(A_{n \times n}\) be a matrix then minor \(M_{ij}\) of the element \(a_{ij} \in A\) is the determinant of the matrix \(B_{(n - 1) \times (n - 1)}\) obtained by deleting the \(i^{\text{th}}\) row and \(j^{\text{th}}\) column from \(A\).

Co-factor of an Element of a Matrix

If \(C_{ij}\) is the cofactor of \(a_{ij}\) element then

\[C_{ij} = (-1)^{i + j} M_{ij}\]

Theorem

\[\det(A_{n \times n})\iff A_{n \times n}^{-1} \quad \text{exists}\]

Theorem

\[A^{-1} = \frac{\text{Adj}(A)}{\det(A)}\]
Adjoint of \(A\)

\(\text{Adj}(A) = [C_{ij}]^T\)

Derivative of a Matrix of Functions2

If \(A(t)\) is a matrix with elements which are functions which are differentiable5 on a common interval6

\[A(t) = [a_{ij}(t)]_{m \times n}\]

then

\[\frac{d}{dt} A(t) = \left[\frac{d a_{ij}}{dt}\right]_{m \times n}\]

It can be denoted as \(A^\prime(t)\).

Integral of a Matrix of Functions2

If \(A(s)\) is a matrix with elements which are functions which are continuous7 on a common interval6 containing \(t\)

\[ \int_{t_0}^t A(s) ds = \left[ \int_{t_0}^t a_{ij}(s)ds \right]_{m \times n} \]

References

Read more about notations and symbols.

  1. Read more about numbers. ↩

  2. Read more about functions. ↩↩↩

  3. Read more about vectors. ↩

  4. Read more about invertible matrices. ↩

  5. Read more about differentiation. ↩

  6. Read more about intervals. ↩↩

  7. Read more about continuous. ↩