Math 11 Matrix Multiplication and Determinant

​A matrix is a block of numbers which includes multiple distinct values. Matrices (the plural of "matrix") have dimensions which are written as the number of rows x the number of columns. Each number within a matrix is called a "term" of that matrix. "Corresponding terms" are terms in different matrices that are in the same location in their respective matrices.

​We can perform operations with matrices. To add two matrices, you add corresponding terms to each other. Subtraction works exactly the same way except you subtract the corresponding terms instead of adding them.

Some matrices cannot be added or subtracted. If the matrices do not have the same dimensions, then they cannot be added to each other.​ The top matrices cannot be added since they have different dimensions. The bottom matrices can be added since they have the same dimensions.

​When we multiply a matrix by a scalar, we multiply each term by the scalar outside the matrix. In the example to the right, we multiply each term in the matrix by 2.

​Multiplying matrices takes more time and steps. First, we have to make sure that we can multiply the two matrices we want to multiply. We have to compare the number of columns in the first matrix to the number of rows in the second matrix. If they match, then we can multiply the matrices. If they do not, then we cannot multiply them and there is no solution.

​The product matrix will have the number of rows from the first multiple and the number of columns from the second multiple. This helps us make sure we are multiplying correctly.

Once we know we can multiply matrices, then we can start the process. First, highlight the first row in the first matrix and the first column in the second matrix. Multiply the first terms, then the second terms, and then the third terms, and so on. Add the products of each multiplication and that is the first term in the product matrix.

After the first term is found, using the first row from the first matrix, do the same operation using the second column in the second matrix. The resulting term is in the first row, second column of the product matrix. Continue this process using the first row until there are no more columns left. Then, repeat the process using the next row.