Basics of Matrix

What is Matrix?

Matrix is a two-dimensional array of numbers. In it numbers are arranged in rows and columns, and they are enclosed by a square bracket, i.e. by [ ].

For example, $\begin{bmatrix} 1 & 2 & 3 \newline a & b & c \end{bmatrix}$

Matrix is singular, while Matrices is plural.

Rows and Columns of a Matrix

The rows are horizontal, i.e. numbers written left to right.
The columns are vertical, i.e. numbers written from top to bottom.

For example, in the matrix A = $\begin{bmatrix} 1 & 2 & 3 \newline 4 & 5 & 6 \end{bmatrix}$:

  • the two horizontal rows are: (1, 2, 3) and (4, 5, 6)
  • the three vertical columns are: (1, 4), (2, 5) and (3, 6)

So, the above matrix is a 2 × 3 matrix (row × column).

Denoting an element of a Matrix

While we use capital letters to denote matrices, we use small letters to denote their elements. An element in a matrix can be denoted using row and column numbers in the subscript. The row number is written first, followed by the column number.

For example, in the above matrix A, $a_{1,2}$ means the element in the first row and second column, which is 2.

Similarly, $a_{2,3}$ means the element in the second row and third column, which is 6.

You may remember it using a mnemonic arc, i.e. $a_{r,c}$. The name of the matrix in small letter, with rows and columns in subscript (in that order).

Properties of Matrices

Addition and Subtraction of Matrices

Two matrices having the same rows and columns can be added or subtracted. Matrix

For example:

$\begin{bmatrix} 1 & 2 \newline 3 & 4 \end{bmatrix}$ + $\begin{bmatrix} 5 & 6 \newline 7 & 8 \end{bmatrix}$ = $\begin{bmatrix} 6 & 8 \newline 10 & 12 \end{bmatrix}$

$\begin{bmatrix} 5 & 6 \newline 7 & 8 \end{bmatrix}$ - $\begin{bmatrix} 1 & 2 \newline 3 & 4 \end{bmatrix}$ = $\begin{bmatrix} 4 & 4 \newline 4 & 4 \end{bmatrix}$

Multiplication of Matrices

We may encounter two types of multiplications when matrices are involved.

Scalar multiplication

We can multiply a matrix by a constant (called scalar). Matrix

For example, 3 × $\begin{bmatrix} 1 & 2 \newline 3 & 4 \end{bmatrix}$ = $\begin{bmatrix} 3 & 6 \newline 9 & 12 \end{bmatrix}$

When we multiply a matrix by -1, we get the negative matrix.

Multiplication of two Matrices

We can also multiply two metrices, say A × B. We will study this in more detail in a separate article.

We cannot divide two matrices in a traditional way. Rather we multiply one matrix with the inverse of the other matrix.

For example, if we have to find out A/B, then we will do it this way:
A/B = A × (1/B) = A × $B^{-1}$
Where $B^{-1}$ is the Inverse of matrix B.

Transposing a Matrix

To transpose a matrix, we just swap the rows and columns.

For example, if we have a matrix A = $\begin{bmatrix} 1 & 2 & 3 \newline 4 & 5 & 6 \end{bmatrix}$, then:

Transpose of matrix A, $A^T$ = $\begin{bmatrix} 1 & 4 \newline 2 & 5 \newline 3 & 6 \end{bmatrix}$

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