Matrices For Class 12
Matrices Class 12 Notes
Matrix is one of the important concepts of Mathematics and one of the most powerful tools, which has various applications such as in solving linear equations, budgeting, sales projection, cost estimation, etc. Matrices for class 12 covers the important concepts in matrices, such as types, order, matrix elementary transformation operations and so on. Students can get a detailed explanation of matrix concepts here. Matrices for class 12 helps the students with their higher studies, as it covers all the basic topics. Go through the notes on class 12 matrices to score good marks in the examinations.
Matrices for Class 12 Topics
The topics covered in matrices for class 12 include the following topics:
- Introduction
- Matrix
- Types of Matrices
- Operations on Matrices
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Elementary Operation (Transformation) of a Matrix
- Invertible Matrices
Matrices Definition
A matrix is a function that consists of an ordered rectangular array of numbers. The numbers in the array are called the entities or the elements of the matrix. The horizontal array of elements in the matrix is called rows, and the vertical array of elements are called the columns. If a matrix has m rows and n columns, then it is known as the matrix of order m x n.
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Types of Matrices
Depending upon the order and elements, matrices are classified as:
Let’s understand the definition of all these types of matrices along with examples here.
Type of matrix | Definition and Example |
Column matrix | A column matrix is an m × 1 matrix, consisting of a single column of m elements. It is also called a column vector.
Example: \(\begin{bmatrix} 4 \\ 1\\ -5 \end{bmatrix}\) |
Row matrix | A row matrix is a 1 × m matrix, consisting of a single row of m elements. It is also called a row vector.
Example: \(\begin{bmatrix} 2 &-1&0\\ \end{bmatrix}\) |
Square matrix | A matrix that has an equal number of rows and columns. It is expressed as m × m.
Example: Square matrix of order 2 is \(\begin{bmatrix} 1 &8 \\ -3 &1 \end{bmatrix}\). Square matrix of order 3 is \(\begin{bmatrix} 1 &-1&-4\\ 8&1&2\\ 0&3&1 \end{bmatrix}\). |
Diagonal matrix | A square matrix that has non-zero elements in its diagonal part running from the upper left to the lower right or vice versa.
Example: \(\begin{bmatrix} 9 &0&0\\ 0&-4&0\\ 0&0&6 \end{bmatrix}\) |
Scalar matrix | The scalar matrix is a square matrix, which has all its diagonal elements equal and all the off-diagonal elements as zero.
Example: \(\begin{bmatrix} \frac{1}{4} &0&0\\ 0&\frac{1}{4}&0\\ 0&0&\frac{1}{4} \end{bmatrix}\) |
Identity matrix | A square matrix that has all its principal diagonal elements as 1’s and all non-diagonal elements as zeros.
Example: Identity (Unit) matrix of order 2 is \(\begin{bmatrix} 1 &0 \\ 0 &1 \end{bmatrix}\). Identity matrix of order 3 is \(\begin{bmatrix} 1 &0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}\). |
Zero matrix | A matrix whose all entries are zero. It is also called a null matrix.
Example: \(\begin{bmatrix} 0 &0&0\\ 0&0&0\\ \end{bmatrix}\) |
Equality of Matrices
Two matrices are said to be equal if-
(i) The order of both the matrices are the same
(ii) Each element of one matrix is equal to the corresponding element of the other matrix
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Operations on Matrices
In Chapter 3 of Class 12 Matrices, certain operations on matrices are discussed, namely, the addition of matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices.
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Transpose of a Matrix
If A = [a_{ij}] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A′ or (A^{T} ).
In other words, if A = [a_{ij}] m × n , then A′ = [a_{ji}] n × m .
Example:
Matrix A = \(\begin{bmatrix} 2 &1&3\\ -4&0&5\\ \end{bmatrix}\)
Transpose of A = A^{T} = \(\begin{bmatrix} 2 &-4\\ 1& 0\\ 3&5\\ \end{bmatrix}\)
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Symmetric and Skew Symmetric Matrices
A square matrix A = [a_{ij}] is said to be symmetric if the transpose of A is equal to A, that is, [a_{ij}] = [a_{ji}] for all possible values of i and j.
A square matrix A = [a_{ij}] is a skew-symmetric matrix if A′ = – A, that is a_{ji} = – a_{ij} for all possible values of i and j. Also, if we substitute i = j, we have a_{ii} = – a_{ii} and thus, 2a_{ii} = 0 or a_{ii} = 0 for all i’s. Therefore, all the diagonal elements of a skew symmetric matrix are zero.
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Elementary Operation (Transformation) of a Matrix
There are six operations (transformations) on a matrix, three of which are due to rows, and three are due to columns, known as elementary operations or transformations.
- The interchange of any two rows or two columns.
- The multiplication of the elements of any row or column by a non zero number.
- The addition to the elements of any row or column, the corresponding elements of any other row or column are multiplied by any non zero number.
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Invertible Matrices
Suppose a square matrix A of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A, and it is denoted by A^{-1}. Also, matrix A is said to be an invertible matrix here.
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Matrices for Class 12 Examples
Example 1:
If \(\begin{bmatrix} x+3 & z+4 & 2y-7\\ -6 & a-1 & 0\\ b-3 & -21 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 6 & 3y-2 \\ -6 & -3 & 2c +2\\ 2b+4 & -21 & 0 \end{bmatrix}\), then find the value of a, b, c, x, y, and z.
Solution:
It is given that, the two matrices are equal. Therefore, the corresponding elements present in matrices should be equal to each other. By comparing the corresponding elements in the matrices, we get:
x+3 = 0.
⇒ x = -3
z +4 = 6
⇒ z = 6-4
⇒ z = 2
2y-7 = 3y-2
⇒3y-2y =-7+2
⇒y = -5
a-1 = -3
⇒a = -3+1
⇒a=-2
2c+2 = 0
⇒2c = -2
⇒ c = -1
b-3 = 2b+4
⇒2b-b = -3-4
⇒ b = -7
Therefore, the values of the variables are:
a = -2
b = -7
c = -1
x = -3
y = -5
z = 2
Example 2:
If \(A=\begin{bmatrix} 1\\ 3\\ -6\\ \end{bmatrix}\) and \(B=\begin{bmatrix} -2&4&5 \end{bmatrix}\), then verify that (AB)^{T} = B^{T}A^{T}.
Solution:
Given,
\(A=\begin{bmatrix} 1\\ 3\\ -6\\ \end{bmatrix}\) and \(B=\begin{bmatrix} -2&4&5 \end{bmatrix}\)
\(AB=\begin{bmatrix} 1\\3\\-6 \end{bmatrix}\begin{bmatrix} -2&4&5 \end{bmatrix}\\=\begin{bmatrix} -2&4&5\\ -6&12&15\\ 12&-24&-30 \end{bmatrix}\)
Now, we need to calculate the transpose of AB.
\((AB)^T=\begin{bmatrix} -2&-6&12\\ 4&12&-24\\ 5&15&-30 \end{bmatrix}\)
\(A^T=\begin{bmatrix} 1&3&-6 \end{bmatrix}\)
And
\(B^T=\begin{bmatrix} -2\\ 4\\5 \end{bmatrix}\)
\(B^TA^T=\begin{bmatrix} -2\\ 4\\5 \end{bmatrix}\begin{bmatrix} 1&3&-6 \end{bmatrix}=\begin{bmatrix} -2&-6&12\\ 4&12&-24\\ 5&15&-30 \end{bmatrix}\)
Therefore, (AB)^{T} = B^{T}A^{T}.
Hence verified.
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