Algebra 2 - Matrix Operations


  • A matrix is an arrangement of numbers in a rectangular array (in rows and columns).
  • Generally, a matrix of the order m×n is symbolically represented as Aijm×n ,where m and n denotes the number of rows and columns.
  • 15-2-237 is an example of a matrix of order 2×3 because it has two rows and three columns.
  • 2-15703-38614-1 is another example of a matrix of order 4×3 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: 31-23×1 is a column matrix of order 3×1 i.e. 3 rows and 1 column.

2. Row Matrix

    • a matrix with only one row
    • Example: 3-1051×4 is a row matrix of order 1×4 i.e. 1 row and 4 columns.

3. Square Matrix

    • a matrix with the same number of rows and columns
    • Example: -1023617-513×3 is a square matrix of order 3×3 i.e. 3 rows and 3 columns.

4. Diagonal Matrix

    • a type of square matrix with all its non-diagonal elements as zero
    • Example: -2000050000-1000094×4 is a diagonal matrix of order 4×4 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: 1000100013×3 is an identity matrix of order 3×3  i.e. 3 rows and 3 columns.

Equality of Matrices

  • Two matrices X & Y can be said to be equal only if -
    1. they are of the same order
    2. 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


  • 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. A+B=B+A, where A and B are matrices of the same order.
  • Matrix addition is associative i.e. A+B+C=A+B+C, where A, B, and C are matrices of the same order.



  • 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. A-BB-A, where A and B are matrices of the same order.
  • Matrix subtraction is non-associative i.e. A-B-CA-B-C, 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 aijm×n,then kA=k·aijm×n where k 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 A=aijm×n is multiplied by another matrix B=bijn×p, then the resultant matrix will be C=cikm×p, where cik=aij·bjkj=1n.
  • If AB and BA are defined, it is not necessary that they are equal as the orders of AB and BA 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 A is deonted by AT or A'.
  • If we interchange the rows and columns of a matrix A=aijm×n then, the resultant matrix obtained AT=ajin×m is called the transpose of matrix A.
  • If A is a m×n matrix, then AT will be a n×m matrix.
  • The transpose of a row matrix is a column matrix and vice versa.

Solved Examples

Example 1: Add the matrices A and B, where A=105-23-1864 and B=39-110237-42

 Solution: A+B=105-23-1864+39-110237-42=1+30+95-1-2+103+2-1+38+76-44+2=4948521526

Example 2: Find the sum of the matrices A and B, where A=2-1532×2 and A=5139-172×3

Solution: The addition of given matrices can't be evaluated as they don't have the same dimensions. Matrix A is of the order 2×2 whereas matrix B is of the order 2×3.

Example 3: Subtract matrix B from matrix A, where A=-31522×2 and B=7-2612×2

Solution: A-B=-31522×2-7-2612×2=-3-71+25-62-12×2=-103-112×2

Example 4: If A=435-12×2, then evaluate 7A.

Solution: 7A=4×73×75×7-1×72×2=282135-72×2. In this example, every element of matrix A is multiplied by 7 to produce the scalar multiple.

Example 5: If X=2-738512×3, then find the value of kX where k=-1.

Solution: kX=2×-1-7×-13×-18×-15×-11×-12×3=-27-3-8-5-12×3

In the example above, where k=-1, 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 X=2-157-343×2 and Y=16425-13×2.

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 A=2-157-343×2 and B=15634-22×3.

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 A=2-314-562×3

Solution: AT=24-3-5163×2. It should be noted that the order of A is 2×3 while the order of AT is 3×2.

Example 9: If A=9-11-26 and B=1-685, then find the value of A+2BT.

Solution: A+2BT=AT+2BT=9-2-116+216-1210=1114-2316

Cheat Sheet

Matrix Addition:

    • 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.

Matrix Subtraction:

    • 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.

Scalar Multiplication:

    • 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.

Blunder Areas

  • 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 AB=BA.
  • It should be kept in mind that ABT=BTAT.