Matrix

Matrices

3.1 Overview

A matrix is an ordered rectangular array of numbers or functions. For example, the matrix $A$ is: A=\begin{bmatrix}x&4&3\\ 4&3&x\\ 3&x&4\end{bmatrix} [cite_start]The numbers or functions within the matrix are called the elements or entries. The horizontal lines of elements form the rows, and the vertical lines form the columns.

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3.1.2 Order of a Matrix

A matrix with $m$ rows and $n$ columns is called a matrix of order $m\times n$ or simply an $m\times n$ matrix. The matrix $A$ above is a $3\times3$ matrix. In general, an $m\times n$ matrix has the following rectangular array:

A=[a_{ij}]_{m \times n}=\begin{bmatrix}a_{11}&a_{12}&\dots&a_{1n}\\ a_{21}&a_{22}&\dots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\dots&a_{mn}\end{bmatrix}

The element $a_{ij}$ is the element in the $i^{th}$ row and $j^{th}$ column. An $m\times n$ matrix contains $mn$ elements.

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3.1.3 Types of Matrices

The document defines several types of matrices:

• Row matrix: a matrix with only one row.
• Column matrix: a matrix with only one column.
• Square matrix: a matrix where the number of rows equals the number of columns, i.e., $m=n$.
• Diagonal matrix: a square matrix where all non-diagonal elements are zero. For a matrix $B=[b_{ij}]_{n\times n}$, this means $b_{ij}=0$ when $i\ne j$.
• Scalar matrix: a diagonal matrix where all diagonal elements are equal to a constant, $k$. This means $b_{ij}=0$ when $i\ne j$ and $b_{ij}=k$ when $i=j$.
• Identity matrix: a square matrix where all diagonal elements are 1 and the rest are all zeroes. A square matrix $A=[a_{ij}]_{n\times n}$ is an identity matrix if $a_{ij}=1$ when $i=j$ and $a_{ij}=0$ when $i\ne j$.
• Zero matrix or null matrix: a matrix where all its elements are zero. It is denoted by O.

Two matrices, $A=[a_{ij}]$ and $B=[b_{ij}]$, are equal if they have the same order and each element of A is equal to the corresponding element of B, i.e., $a_{ij}=b_{ij}$ for all $i$ and $j$.

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3.1.4 Addition of Matrices

Two matrices can be added only if they have the same order.

3.1.5 Multiplication of Matrix by a Scalar

If $A=[a_{ij}]_{m\times n}$ is a matrix and $k$ is a scalar, then the matrix $kA$ is obtained by multiplying each element of A by the scalar $k$, i.e., $kA=[ka_{ij}]_{m\times n}$.

3.1.6 Negative of a Matrix

The negative of a matrix A is denoted by $-A$ and is defined as $(-1)A$.

3.1.7 Multiplication of Matrices

The multiplication of two matrices, A and B, is defined only if the number of columns in A is equal to the number of rows in B. If $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, their product is a matrix C of order $m\times p$. The element $c_{ik}$ of matrix C is obtained by multiplying the elements of the $i^{th}$ row of A with the corresponding elements of the $k^{th}$ column of B and summing the products.

Important notes on matrix multiplication:

• If the product $AB$ is defined, the product $BA$ is not necessarily defined.
• If both $AB$ and $BA$ are defined, they are not necessarily equal ($AB\ne BA$).
• The product of two matrices can be a zero matrix even if neither of the original matrices is a zero matrix.
• For three matrices A, B, and C of the same order, if $A=B$, then $AC=BC$, but the converse is not always true.

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3.1.8 Transpose of a Matrix

The transpose of an $m\times n$ matrix A, denoted by $A^{\prime}$ or $(A^{T})$, is the matrix obtained by interchanging the rows and columns of A. If $A=[a_{ij}]_{m\times n}$, then its transpose is $A^{T}=[a_{ji}]_{n\times m}$.

Properties of a matrix transpose for matrices A and B of suitable orders:

• $(A^{T})^{T}=A$.
• $(kA)^{T}=kA^{T}$, where $k$ is any constant.
• $(A+B)^{T}=A^{T}+B^{T}$.
• $(AB)^{T}=B^{T}A^{T}$.

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3.1.9 Symmetric Matrix and Skew Symmetric Matrix

• A square matrix $A=[a_{ij}]$ is symmetric if $A^{T}=A$, which means $a_{ij}=a_{ji}$ for all $i$ and $j$.
• A square matrix $A=[a_{ij}]$ is skew symmetric if $A^{T}=-A$, which means $a_{ij}=-a_{ji}$ for all $i$ and $j$. The diagonal elements of a skew symmetric matrix are always zero.

Theorems:

• For any square matrix A with real number entries, $A+A^{T}$ is a symmetric matrix, and $A-A^{T}$ is a skew symmetric matrix.
• Any square matrix A can be expressed as the sum of a symmetric and a skew symmetric matrix: $A=\frac{(A+A^{T})}{2}+\frac{(A-A^{T})}{2}$.

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3.1.10 Invertible Matrices

A square matrix A of order $m\times m$ is invertible if there exists another square matrix B of the same order such that $AB=BA=I_{m}$. Matrix B is called the inverse of A and is denoted by $A^{-1}$.

Important notes on invertible matrices:

• A rectangular matrix does not have an inverse.
• If B is the inverse of A, then A is also the inverse of B.
• The inverse of a square matrix, if it exists, is unique.
• If A and B are invertible matrices of the same order, then $(AB)^{-1}=B^{-1}A^{-1}$.

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3.1.11 Inverse of a Matrix using Elementary Row or Column Operations

To find $A^{-1}$ using elementary row operations, you write $A=IA$ and apply a sequence of row operations until you get the form $I=BA$. The matrix B is then the inverse of A. Similarly, for column operations, you start with $A=AI$ and apply a sequence of column operations until you get $I=AB$. If, at any point, all elements in one or more rows (or columns) of the matrix on the left-hand side become zero, the inverse does not exist. [^1]

References

[^1] : "Matrices" from Maths.scot