FOR GMS ICSE 2014 7

Multiplication of Matrices

In this class we wish to work at multiplying two matrices. This is different from multiplying a matrix by a real number.

When can two matrices be multiplied ? Two matrices can be multiplied ONLY if the number of columns in the first matrix equals the number of rows in the second matrix. 

• Thus if A is a 2 x 3 matrix and B is a 3 x 2 matrix, we can multiply the matrices - why ? because the number of columns in A (3) is equal to the number of rows in B (3).
• But if X is a 2 x 3 matrix and Y is a 2 x 3 matrix matrix, then we cannot multiply the matrices. Why ? Because the number of columns in matrix X (3) is not equal to the number of rows (2) in matrix Y.
• Please note the matrix we get will have the number of rows of the first matrix and the number of columns of the second matrix. In this case the answer will be a 2 x 2 matrix. Please make sure you understand this.

Now please refer to page 167and page 168 : 12.13 in your text books. The multiplication of matrices is given clearly - it shows the multiplication of two 2 x 2 matrices.

• the matrices can be multiplied because the columns of the first (2) are equal to the rows of the second (2).
• How many rows and columns will the answer have ? : 2 x 2 
• How many elements does a 2 x 2 matrix have ? 4
• Therefore the number of steps will be 4
• as shown on page 167 / 168

Now let's look at Ex.12(C) 1 (iii) :

• we asked if we can multiply a 2 x 2 matrix with 2 x 1 matrix.
• Can we do so ? Yes because the number of columns of the first matrix (2) are equal to the number of rows in the second matrix (2).
• What will be the order (rows and columns) of the matrix we derive ? a 2 x 1 matrix
• How many elements are there in a 2 x 1 matrix ? 2
• Step 1 : (6 x -1) + (4 x 3) = -6 + 12 = 6
• Step 2 : (3 x -1) + (-1 x 3) = -3 + (-3) = -3 -3 = -6
• Then you write the matrix
• Do you follow the process ?
• Then please do Ex 12 (C) 1 and 2 Please remember that in 2 (iii) and (iv) I is an Identity or Unit matrix - please look at pg.160 7 to refresh your memory.

All the best !