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Matrices is based on the various properties of matrices, Invertible Matrices, elementary row and column operations in matrices, Transpose of a Matrix and identity matrix. The number or functions are referred to as the elements or the entries of the matrix. Matrices can be divided into different types which is categorized based on their order, number of rows and columns, value of their elements, etc. This is important as the concept of matrices will be useful in different domains such as sales, business, cost estimation and so on.
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Keyterms: Matrices, Invertible Matrices, Transpose of a Matrix, Identity matrix, Row matrix, Column matrix, Zero, Symmetric, Skew Symmetric matrix
Define Matrix
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Matrix is a rectangular table or array of symbols, numbers or expressions, arranged in columns and rows. The dimension of a matrix is 2x3 or below, as there are two rows and three columns.
With the use of straight-forward methods, Matrix simplifies the complex work and breaks down the information; which is why it is deemed as one of the most powerful tools in mathematics. Matrix has evolved through constant attempts to extract simple and compact methods of solving systems of linear equations. Areas of business such as sales projection, budgeting, analysis, and cost estimation use the operations of notations of the matrix. Matrix also transcends through our screen and is used in physical operations such as rotation, reflection, and magnification. Along with this, Matrix is also used in cryptography, genetics, economics, modern psychology, sociology, and industrial management.
Types of Matrices
Matrices can be classified into many broad categories, based on the number of rows, columns, and many other factors.
The main types of matrices
The video below explains this:
Matrices Detailed Video Explanation:
Here are the main types of matrices, along with their explanations, examples and expansions:
| Types of Matrix | Explanation | Expansion |
|---|---|---|
| Row Matrix | A matrix with a single row is considered to be a row matrix. In cases where m = 1, A = [aij]mxn. Row matrix has only one row and the order of a row matrix is 1 x n. Example: In the row matrix of order 1 x 4, A = [ 1 2 4 5] In the row matrix of order 1 x 3, P = [ -4 -21 -17] | A = [aij]1×n |
| Column Matrix | A matrix with one column is considered a column matrix. In A = [aij]mxn, the column matrix is n = 1  | A = [aij]m×1 |
| Zero or Null Matrix | A zero matrix is where all entries are zero or null. For example, [00] is a zero matrix of order 1 x 2. 0 2x2 = \(\begin{bmatrix}0 & 0 \\[0.3em]0 & 0 \\[0.3em] \end{bmatrix}\) 0 3x3 = \(\begin{bmatrix}0 & 0 & 0 \\[0.3em]0 & 0 & 0\\[0.3em]0 & 0 & 0\\[0.3em] \end{bmatrix}\) 0 3x2 = \(\begin{bmatrix}0 & 0\\[0.3em]0 & 0 \\[0.3em]0 & 0 \\[0.3em] \end{bmatrix}\) 0 1x4 = \(\begin{bmatrix}0 & 0& 0& 0\\[0.3em]\end{bmatrix}\) | A = [aij]mxn Here, aij = 0 |
| Horizontal Matrix | A matrix in which columns outnumber the rows is called a horizontal matrix | A = [aij]mxn Here, n > m |
| Vertical Matrix | A matrix in which rows outnumber the columns is called a vertical matrix | A = [aij]mxn Here, m > n |
| Square Matrix | A matrix with the number of rows and columns is equal. A m x n matrix will be square if m = n, and it will be known as the square matrix of order ‘n’. For example: The square matrix of 3 is: A = \(\begin{bmatrix}3 & -1 & 0 \\[0.3em]3/2 & \sqrt{3/2} & 1\\[0.3em]4 & 3 & -1\\[0.3em] \end{bmatrix}\) | A = [aij]mxn Here, m = n |
| Diagonal Matrix | A square matrix is considered a diagonal matrix when its non-diagonal elements are zero. For example, B =[bij]m×m will be diagonal if bij = 0, when i ≠ j. | A = [aij]mxn Here, i ≠ j |
| Scalar Matrix | A diagonal matrix where all principal elements are equal to some non-zero matrix is a scalar matrix. A square matrix B = [bij]nxn is considered to be scaler if: bij = 0, when i ≠ j bij = k, when i = j, for some constant k A = [4] \(\begin{bmatrix}-1 & 0 \\[0.3em]0 & -1 \\[0.3em] \end{bmatrix}\)\(\begin{bmatrix}3 & 0 & 0 \\[0.3em]0 & 3 & 0 \\[0.3em]0 & 0 & 3 \\[0.3em] \end{bmatrix}\) | A = [aij]mxn where, aij = {0, I = j} {k, I ≠ j} Here K is constant |
| Identity Matrix (Unit Matrix) | A square matrix in which all elements are 0 and each diagonal element is non-zero, is called an identity matrix. It is marked by I. | A = [aij]m×n where, aij= {1, i=j {1, i≠j |
| Equal Matrix | Matrices of the same order where their corresponding elements are equal to a square matrix is called an equal matrix. | A = [aij]mxn & B = [bij]rxs here, aij = bij, m = r, & n = s |
| Lower Triangular Matrices | A lower triangular matrix is in which all upper triangular elements are zero. | (aij = 0 when i < j) |
| Upper Triangular Matrix | A square matrix in which all entries are below the main diagonal are zero is called the upper triangular matrix. | aij = 0, when i > j) |
| Singular Matrix | A matrix where there is only one element is considered to be a singular or singleton matrix. For example, A = [aij]mxn is a singular matrix if m = n = 1. | |A| = 0 |
| Non-Singular Matrix | |A| ≠ 0 | |
| Symmetric Matrix | A square matrix is considered to be symmetric when aij = aji for all i and j, in the case in which aij is present at (j,i)th position. \(\begin{bmatrix}1 & 3 & 8 \\[0.3em]3 & 8 & -4 \\[0.3em]8 & -4 & 6 \\[0.3em] \end{bmatrix}\) | A = [aij] (Here, aij = aji) |
| Symmetric and Skew Symmetric Matrices | A square matrix is supposed to be skew-symmetric if aij =−aji for all i and j. In a skew-symmetric matrix, the transpose of matrix A equals the negative of matrix A i.e (AT =−A). Note: All primary diagonal elements in the skew-symmetric matrix are zero. \(\begin{bmatrix}0 & -6 & 4 \\[0.3em]-6 & 0 & 7 \\[0.3em]4 & -7 & 0 \\[0.3em] \end{bmatrix}\) | A = [aij] (Here, aij = aji) |
| Hermitian Matrix | A square matrix is considered to be a hermitian matrix when every diagonal element of the matrix is real. For example, A = [aij] will be hermitian if: \(aij = aij = \overline{a}ji \forall i, j; i.e. A = A^ {\theta}\) | A = Aθ |
| Skew – Hermitian Matrix | A square matrix is considered to be a skew-hermitian if \(aij = - \overline{a}ji, \forall i, j;\) | Aθ = -A |
| Orthogonal Matrix | A square matrix is said to be an orthogonal matrix if the transpose of a matrix is equal to its inverse matrix. | A AT = In = AT A |
| Idempotent Matrix | When A2 = A, a square matrix is considered to be an idempotent matrix. A, An = A∀n > 2, n ∈ N⇒An =A, n≥2. For a matrix to be idempotent, det A = 0 or x. | A² = A |
| Periodic Matrix | A matrix in which the square matrix satisfies the relation Ak+1 = A, for positive integer K, and A is periodic with period K. If K is the least positive integer for which Ak+1 =A | Ak+1 = A |
| Involutory Matrix | A matrix in which A 2 = I is called an involuntary matrix. An involutory matrix is its own inverse. For example, E.g. (i) A = \(\begin{bmatrix}0 & 1 \\[0.3em]1 & 0 \\[0.3em] \end{bmatrix}\)\(\begin{bmatrix}0 & 1 \\[0.3em]1 & 0 \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix}1 & 0 \\[0.3em]0 & 1 \\[0.3em] \end{bmatrix}\) | A2 = I, A-1 = A |
| Nilpotent Matrix | A matrix that is nilpotent of index p is called nilpotent matrix. If p is the least positive, then (p∈N),ifA p=O,A p−1=O, Here Ap = O. | ∃ p ∈ N (Given, AP = 0) |
Also Read:
Points to Remember
- Matrices is based on the various properties of matrices, invertible matrices, elementary row and column operations in matrices, transpose of a matrix and identity matrix.
- The number or functions are referred to as the elements or the entries of the matrix.
- Matrices can be divided into different types which is categorized based on their order, number of rows and columns, value of their elements, etc.
- Matrix is a rectangular table or array of symbols, numbers or expressions, arranged in columns and rows.
- The dimension of a matrix is 2x3 or below, as there are two rows and three columns.
- Matrices can be classified into many broad categories, based on the number of rows, columns, and many other factors.
Also Read:
Sample Questions
Objective Type Questions
Ques. If 2 \(\begin{bmatrix}3 & 4 \\[0.3em]5 & x \\[0.3em] \end{bmatrix} + \begin{bmatrix}1 & y \\[0.3em]0 & 1 \\[0.3em] \end{bmatrix}= \begin{bmatrix}7 & 0 \\[0.3em]10 & 5 \\[0.3em] \end{bmatrix}\), then find (x – y). (Delhi 2014)
Ans. Given, 2 \(\begin{bmatrix}3 & 4 \\[0.3em]5 & x \\[0.3em] \end{bmatrix} + \begin{bmatrix}1 & y \\[0.3em]0 & 1 \\[0.3em] \end{bmatrix}= \begin{bmatrix}7 & 0 \\[0.3em]10 & 5 \\[0.3em] \end{bmatrix}\)
⇒ \(\begin{bmatrix}6 & 8 \\[0.3em]10 & 2x \\[0.3em] \end{bmatrix} + \begin{bmatrix}1 & y \\[0.3em]0 & 1 \\[0.3em] \end{bmatrix}= \begin{bmatrix}7 & 0 \\[0.3em]10 & 5 \\[0.3em] \end{bmatrix}\)
⇒\(\begin{bmatrix}7 & 8+y \\[0.3em]10 & 2x+1 \\[0.3em] \end{bmatrix} = \begin{bmatrix}7 & 0 \\[0.3em]10 & 5 \\[0.3em] \end{bmatrix}\)
By comparing the corresponding elements, we get,
8 + y = 0 and 2x + 1 = 5
⇒ y = – 8 and x = \(\frac{5-1}{2}\)= 2
∴ x- y = 2 – (- 8) = 10
Ques. Solve the following matrix equation for x. [x 1] \(\begin{bmatrix}1 & 0 \\[0.3em]- 2 & 0 \\[0.3em] \end{bmatrix}\)= 0 (Delhi 2014)
Ans. We have, [x 1]\(\begin{bmatrix}1 & 0 \\[0.3em]- 2 & 0 \\[0.3em] \end{bmatrix}\)= 0
By using matrix multiplication, we get [x – 2 0] = [0 0]
By comparing the corresponding elements from both the sides, we get
x – 2 = 0 ⇒ x = 2
Ques. If A is a square matrix such that A2 = A, then find out the value of 7A – (I –+ A)3, where I is an identity matrix. (AI 2014)
Ans. We have A2 = A
Now, 7A – (I + A)3= 7A – [I3 + A3 + 3IA (I + A)]
[∴ (x + y)3 = x3 + y3 + 3xy (x + y)]
= 7A – [(I + A)2 . A + 3A (I + A)] [\(\because\)I3 = I]
= [(I + A) . A + 3AI + 3A2)] [\(\because\) A2 = A, given]
= [I + A + 3A + 3A)] [\(\because\) AI = A]
= 7A – [I + 7A] = – I (1)
Ques. Find the value of a if,? \(\begin{bmatrix}a-b & 2a + c \\[0.3em]2a-b &3c + d \\[0.3em] \end{bmatrix} = \begin{bmatrix}-1 & 5 \\[0.3em]0 & 13 \\[0.3em] \end{bmatrix}\)(Delhi 2013)
Ans. It is already known that two matrices are equal if their corresponding elements are equal.
∴ a – b = – 1 …(i)
and 2a – b = 0 …(ii)
On substracting Eq. (i) from Eq. (ii), We get
a = 1 (1/2)
Ques.What is a column matrix? (1 mark)
Ans. The matrix that has only one column is known as column matrix.
In A = [aij]mxn, the column matrix is n = 1
Ques. What do you mean by scalar matrix? (1 mark)
Ans. A diagonal matrix where all principal elements are equal to some non-zero matrix is a scalar matrix.
A square matrix B = [bij]nxn is considered to be scaler if:
- bij = 0, when i ≠ j
- bij = k, when i = j, for some constant k
Ques. How are skew-symmetric matrices different from symmetric matrices? (1 mark)
Ans. A square matrix is supposed to be skew-symmetric if aij =−aji for all i and j.
In a skew-symmetric matrix, the transpose of matrix A equals the negative of matrix A i.e (AT =−A). Here all primary diagonal elements in the skew-symmetric matrix are zero.
Whereas, A square matrix is considered to be symmetric when aij = aji for all i and j, in the case in which aij is present at (j,i)th position.
Ques. Define vertical and horizontal matrices. (1 mark)
Ans. A matrix in which rows outnumber the columns is called a vertical matrix where, m > n.
On the other hand, a matrix in which columns outnumber the rows is called a horizontal matrix where, n > m.
Descriptive Type Questions
Ques. If A = \(\begin{bmatrix}2 & 0 & 1\\[0.3em]2 & 1 & 3 \\[0.3em]1 & -1 & 0 \\[0.3em] \end{bmatrix}\) , then find the value of A2 – 3A = 2I. (All India 2010)
Ans. Given,
A = \(\begin{bmatrix}2 & 0 & 1\\[0.3em]2 & 1 & 3 \\[0.3em]1 & -1 & 0 \\[0.3em] \end{bmatrix}\)
And we have to find out the value of A2 – 3A = 2I.
Now, A2 = A.A
= \(\begin{bmatrix}2 & 0 & 1\\[0.3em]2 & 1 & 3 \\[0.3em]1 & -1 & 0 \\[0.3em] \end{bmatrix}\)\(\begin{bmatrix}2 & 0 & 1\\[0.3em]2 & 1 & 3 \\[0.3em]1 & -1 & 0 \\[0.3em] \end{bmatrix}\)
= \(\begin{bmatrix}4+0+1 & 0+0-1 & 2+0+0\\[0.3em]4+2+3 & 0+1-3 & 2+3+0 \\[0.3em]2-2+0 & 0-1-0 & 1-3+0 \\[0.3em] \end{bmatrix}\)
[multiplying row by column]
⇒ A2 = \(\begin{bmatrix}5 & -1 & 2\\[0.3em]9 & -2 & 5 \\[0.3em]0 & -1 & -2 \\[0.3em] \end{bmatrix}\) (1½ )
3A = 3\(\begin{bmatrix}2 & 0 & 1\\[0.3em]2 & 1 & 3 \\[0.3em]1 & -1 & 0 \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix}6 & 0 & 3\\[0.3em]6 & 3 & 9 \\[0.3em]3 & -3 & 0 \\[0.3em] \end{bmatrix}\) (1/2)
and 2I = 2 \(\begin{bmatrix}1 & 0 & 0\\[0.3em]0 & 1 & 0 \\[0.3em]0 &0 & 1 \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix}2 & 0 & 0\\[0.3em]0 & 2 & 0 \\[0.3em]0 &0 & 2 \\[0.3em] \end{bmatrix}\) (1/2)
∴ A2 – 3A + 2I
∴ A2 – 3A + 2I
= \(\begin{bmatrix}5 & -1 & 2\\[0.3em]9 & -2 & 5 \\[0.3em]0 & -1 & -2 \\[0.3em] \end{bmatrix}\)– \(\begin{bmatrix}6 & 0 & 3\\[0.3em]6 & 3 & 9 \\[0.3em]3 & -3 & 0 \\[0.3em] \end{bmatrix}\)+ \(\begin{bmatrix}2 & 0 & 0\\[0.3em]0 & 2 & 0 \\[0.3em]0 &0 & 2 \\[0.3em] \end{bmatrix}\)
⇒A2 – 3A + 2I
= \(\begin{bmatrix}5-6+2 & -1-0+0 & 2-3+0\\[0.3em]9-6+0 & -2-3+2 & 5-9+0 \\[0.3em]0-3+0 & -1+3+0 & -2-0+2 \\[0.3em] \end{bmatrix}\)
⇒A2 – 3A + 2I = \(\begin{bmatrix}1 & -1 & -1\\[0.3em]3 & -3 & -4 \\[0.3em]-3 &2 & 0 \\[0.3em] \end{bmatrix}\) (1½ )
Ques. Prove that A2 – 4A – 5I = 0, if A = \(\begin{bmatrix}1 & 2 & 2\\[0.3em]2 & 1 & 2 \\[0.3em]2 &2 & 1 \\[0.3em] \end{bmatrix}\) (Delhi 2008)
Ans. Given, A = \(\begin{bmatrix}1 & 2 & 2\\[0.3em]2 & 1 & 2 \\[0.3em]2 &2 & 1 \\[0.3em] \end{bmatrix}\)
∴ A2 = A.A = \(\begin{bmatrix}1 & 2 & 2\\[0.3em]2 & 1 & 2 \\[0.3em]2 &2 & 1 \\[0.3em] \end{bmatrix}\)\(\begin{bmatrix}1 & 2 & 2\\[0.3em]2 & 1 & 2 \\[0.3em]2 &2 & 1 \\[0.3em] \end{bmatrix}\)
= \(\begin{bmatrix}1+4+4 & 2+2+4 & 2+4+2\\[0.3em]2+2+4 & 4+1+4 & 4+2+2 \\[0.3em]2+4+2 & 4+2+2 & 4+4+1 \\[0.3em] \end{bmatrix}\)
= \(\begin{bmatrix}9 & 8 & 8\\[0.3em]8 & 9 & 8 \\[0.3em]8 &8& 9 \\[0.3em] \end{bmatrix}\) (1½ )
Now, LHS = A2 – 4A – 5I
= \(\begin{bmatrix}9 & 8 & 8\\[0.3em]8 & 9 & 8 \\[0.3em]8 &8& 9 \\[0.3em] \end{bmatrix}\)– 4 \(\begin{bmatrix}1 & 2 & 2\\[0.3em]2 & 1 & 2 \\[0.3em]2 &2 & 1 \\[0.3em] \end{bmatrix}\)– 5 \(\begin{bmatrix}1 & 0 & 0\\[0.3em]0 & 1 & 0 \\[0.3em]0 &0 & 1 \\[0.3em] \end{bmatrix}\) (1)
= \(\begin{bmatrix}9 & 8 & 8\\[0.3em]8 & 9 & 8 \\[0.3em]8 &8& 9 \\[0.3em] \end{bmatrix}\) + \(\begin{bmatrix}-4 & -8 &-8\\[0.3em]-8 & -4 & -8 \\[0.3em]-8 &-8 & -4 \\[0.3em] \end{bmatrix}\)+ \(\begin{bmatrix}-5 & 0 &0 \\[0.3em]0 &-5 & 0 \\[0.3em]0 &0 & -5 \\[0.3em] \end{bmatrix}\)
= \(\begin{bmatrix}0 & 0 & 0\\[0.3em]0 & 0 & 0 \\[0.3em]0 &0& 0 \\[0.3em] \end{bmatrix}\) = 0 = RHS (1½ ) Hence Proved.
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