- A matrix is an arrangement of numbers in a rectangular array (in rows and columns).
- Generally, a matrix of the order is symbolically represented as ,where and denotes the number of rows and columns.
- is an example of a matrix of order because it has two rows and three columns.
- is another example of a matrix of order because it has four rows and three columns.
- The individual items in a matrix are called its elements.
Types of Matrices
Some important types of matrices worth remembering are explained below.
1. Column Matrix
- a matrix with only one column
- Example: is a column matrix of order i.e. 3 rows and 1 column.
2. Row Matrix
- a matrix with only one row
- Example: is a row matrix of order i.e. 1 row and 4 columns.
3. Square Matrix
- a matrix with the same number of rows and columns
- Example: is a square matrix of order i.e. 3 rows and 3 columns.
4. Diagonal Matrix
- a type of square matrix with all its non-diagonal elements as zero
- Example: is a diagonal matrix of order i.e. 4 rows and 4 columns in which all of its non-diagonal elements are zero.
5. Identity Matrix
- a type of square matrix in which all the diagonal elements are 1 and all the non-diagonal elements are zero.
- Example: is an identity matrix of order i.e. 3 rows and 3 columns.
Equality of Matrices
- Two matrices X & Y can be said to be equal only if -
- they are of the same order
- each element of matrix X is equal to the corresponding element of matrix Y
Operations on Matrices
Operations on matrices include:
- Addition and Subtraction of matrices
- Multiplication of a matrix by
- a number (scalar multiplication)
- another matrix
- Transpose of a matrix
Note: A matrix can be divided by a scalar (number), but there is no defined rule for dividing a matrix by another matrix.
Addition and Subtraction of Matrices
ADDITION OF MATRICES
- Two matrices can be added or subtracted only if they have the same dimension i.e. they must have the same number of rows and columns.
- To add two matrices, we simply add the corresponding elements of both matrices.
- Matrix addition is commutative i.e. , where A and B are matrices of the same order.
- Matrix addition is associative i.e. , where A, B, and C are matrices of the same order.
SUBTRACTION OF MATRICES
- Similarly to subtract two matrices, we simply subtract the elements of one matrix from the corresponding elements of the other matrix.
- Matrix subtraction is non-commutative i.e. , where A and B are matrices of the same order.
- Matrix subtraction is non-associative i.e. , where A, B, and C are matrices of the same order.
Multiplying a Matrix with a number [Scalar Multiplication]
- When we multiply a matrix by a number, we simply multiply each element of the matrix by that same number. This operation produces a new matrix, which is called a scalar multiple.
- In other words, if matrix A is represented by ,then where is a scalar (number).
Multiplying a Matrix by another Matrix [Matrix Multiplication]
Two matrices A and B can only be multiplied if the number of columns in A is equal to the number of rows in B otherwise not.
- If a matrix is multiplied by another matrix , then the resultant matrix will be , where .
- If and are defined, it is not necessary that they are equal as the orders of and may be different.
- In order to understand the concept of matrix multiplication, refer to Example 7.
Transpose of a Matrix
- The transpose of a matrix is deonted by or .
- If we interchange the rows and columns of a matrix then, the resultant matrix obtained is called the transpose of matrix A.
- If is a matrix, then will be a matrix.
- The transpose of a row matrix is a column matrix and vice versa.
Example 1: Add the matrices A and B, where and
Example 2: Find the sum of the matrices A and B, where and
Solution: The addition of given matrices can't be evaluated as they don't have the same dimensions. Matrix A is of the order whereas matrix B is of the order .
Example 3: Subtract matrix B from matrix A, where and
Example 4: If , then evaluate .
Solution: . In this example, every element of matrix A is multiplied by 7 to produce the scalar multiple.
Example 5: If , then find the value of where .
In the example above, where , the resulting matrix obtained after scalar multiplication is called the negative of the given matrix.
Example 6: Find the product of two matrices X and Y, where and .
Solution: The product can't be evaluated since the number of columns in X (2 here) is NOT equal to the number of rows in Y (3 here).
Example 7: Find the product of two matrices A and B where and .
Solution: In this case, the number of columns in A (2 here) is equal to the number of rows in B (2 here), and hence product can be evaluated.
Example 8: Find the transpose of a matrix
Solution: . It should be noted that the order of is while the order of is .
Example 9: If and , then find the value of .
- Two matrices to be added must be of the same order (size).
- Each element of one matrix is added to the corresponding elements of the other matrix.
- It follows commutative as well as associative laws.
- Two matrices to be subtracted must be of the same order (size).
- Each element of one matrix is subtracted from the corresponding parts of the other matrix.
- It does not obey commutative and associative laws.
- Each matrix element is multiplied by the same scalar (number) to obtain a scalar multiple.
Matrix multiplication (multiplying a matrix by another matrix):
- The product of two matrices, A and B, is only possible when the number of columns of A is equal to the number of rows of B; otherwise, not.
- In general, it is not commutative.
- It obeys associative and distributive laws.
Transpose of a Matrix:
- It is the matrix obtained after interchanging the rows and columns of a given matrix.
- Any square matrix can be represented as the sum of a symmetric and a skew-symmetric matrix.
- If the product of two matrices AB is defined, then BA need not be defined.
- If AB and BA are defined, then it is not necessarily true that .
- It should be kept in mind that .
- Abhishek Tiwari
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