Multiplying Matrices

Definition: When the number of columns in one matrix is equal to the number of rows in another matrix, they can be multiplied.

For example, if Matrix X is 2×3 and Matrix Y is 3×3, they can be multiplied. This is because the number of columns in Matrix X is equal to the number of rows in Matrix Y.

To multiply matrices, we must multiply all rows by all columns and add the products for each:
matrixvectormultiplication.png

Notice that the new matrix has taken on the row dimensions of the first matrix, and the column dimensions of the second matrix.

Example: Multiply the following matrices if possible:

gif.latex-%5Cdpi%7B120%7D %5Cbg_white %5CLARGE A%3D%5Cbegin%7Bbmatrix%7D 1%262 %263 %5C%5C 4%26 5 %266 %5Cend%7Bbmatrix%7D B%3D%5Cbegin%7Bbmatrix%7D 6 %26 5%5C%5C 4 %26 3%5C%5C 2 %26 1 %5Cend%7Bbmatrix%7D.gif

Solution:

Step 1: Check in ensure the number of columns in Matrix A is equal to the number of rows in Matrix B.

Matrix A has 3 columns and Matrix B has 3 rows, so they can be multiplied.

Step 2: Starting with the the first row in Matrix A, and the first column in Matrix B, multiply each row entry by the respective column entry and add their products.

Note that this sum will be placed in the row location from Matrix A (row 1), and the column location from Matrix B (column 1), or ab₁₁

gif.latex-%5Cdpi%7B120%7D %5Cbg_white %5CLARGE %5Cbegin%7Bbmatrix%7D %281%5Ccdot 6%29+%282%5Ccdot 4%29+%283%5Ccdot 2%29%26%26%26%26%26%26%5C%5C %26 %26 %5Cend%7Bbmatrix%7D.gif

Step 3: Do the same with the remaining rows and columns:

gif.latex-%5Cdpi%7B120%7D %5Cbg_white %5CLARGE %5Cbegin%7Bbmatrix%7D %281%5Ccdot 6%29+%282%5Ccdot 4%29+%283%5Ccdot 2%29%26%26%26%26%281%5Ccdot 5%29+%282%5Ccdot 3%29+%283%5Ccdot 1%29%5C%5C %284%5Ccdot 6%29+%285%5Ccdot 4%29&plu.gif

Step 4: Simplify to find the final product of Matrix A and Matrix B:

gif.latex-%5Cdpi%7B120%7D %5Cbg_white %5CLARGE AB%3D%5Cbegin%7Bbmatrix%7D 20%2614%5C%5C 56%2641%5C%5C%5Cend%7Bbmatrix%7D.gif

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