Mathematics: Algebra: Matrices

What is a Matrix:


A matrix is an arrangement of the numbers or functions in the form of a rectangular array. The numbers or functions of a matrix are generally called elements. The matrices are used for writing linear equations in a compact form and solve the equations easily.

Let a girl "A" has 5 pens, then we can write it in the form of a matrix as [5].

If the same girl "A" has 5 pens and 3 pencils, then we can write it in matrix form as [5 3].

Now let if girl "A" has 5 pens, 3 pencils and another girl "B" has 6 pens, 4 pencils, then we can write it in matrix form as $$ \begin{bmatrix} 5 & 3 \\ 6 & 4 \\ \end{bmatrix} $$

Similarly, if girl "A" has 5 pens, 3 pencils, and 2 books. The girl "B" has 6 pens, 4 pencils, and 3 books. The girl "C" has 8 pens, 7 pencils, and 5 books, then we can write it in matrix form as $$ \begin{bmatrix} 5 & 3 & 2 \\ 6 & 4 & 3 \\ 8 & 7 & 5 \\ \end{bmatrix} $$


Order of Matrix:


A matrix with "m" number of rows and "n" number of columns is called the order of \((m \times n)\) matrix.


Notation of Matrix:


$$ A = \begin{bmatrix} 1 & 2 \\ \end{bmatrix} $$ It is a \((1 \times 2)\) matrix or it can be written as \(A_{1 \times 2}\)

$$ B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} $$ It is a \((2 \times 2)\) matrix or it can be written as \(B_{2 \times 2}\)

$$ C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix} $$ It is a \((2 \times 3)\) matrix or it can be written as \(C_{2 \times 3}\)

$$ D = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \\ \end{bmatrix} $$ It is a \((3 \times 2)\) matrix or it can be written as \(D_{3 \times 2}\)

$$ E = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} $$ It is a \((3 \times 3)\) matrix or it can be written as \(E_{3 \times 3}\)


Elements of a Matrix:


$$ \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1j} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2j} & \cdots & a_{2n} \\ a_{i1} & a_{i2} & \cdots & a_{ij} & \cdots & a_{in} \\ \vdots & \vdots& \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mj} & \cdots & a_{mn} \end{pmatrix} $$ This is a \((m \times n)\) matrix. Here \(a_{11}, a_{12}......a_{mn}\) are called elements of matrix. Here \(a_{ij}\) is a general element which is placed at \(i^{th}\) row and \(j^{th}\) column. In the lower suffix of any element, the first number represents the row, and the second number represents the column of the matrix.


Note: Any point \((x, y)\) can also be written in the form of a matrix as \(\begin{bmatrix} x & y \\ \end{bmatrix}\) or \(\begin{bmatrix} x \\ y \\ \end{bmatrix}\)


For example, the point \((3, 2)\) can be written in the form of a matrix as \(\begin{bmatrix} 3 & 2 \\ \end{bmatrix}\) or \(\begin{bmatrix} 3 \\ 2 \\ \end{bmatrix}\)


Example: Read the following information regarding the number of boys and girls students studying in three schools A, B, and C.


$$ \begin{matrix} & Boys & Girls \\ School \ A & 30 & 35 \\ School \ B & 45 & 25 \\ School \ C & 25 & 28 \\ \end{matrix} $$ Write the above information in the form of a matrix, and find what does the element in the second row and the second column represent?


Solution: The matrix form of the given information. $$ A = \begin{bmatrix} 30 & 35 \\ 45 & 25 \\ 25 & 28 \\ \end{bmatrix} $$ It is a \((3 \times 2)\) matrix, it can also be written as \(A_{3 \times 2}\). The element in the second row and second column represents 25 girls students in school B.