To multiply a matrix by a number, multiply each element of the matrix by that number. For example, to multiply a 2x2 matrix by 2, multiply each element by 2. Similarly, to multiply by 3, multiply each element by 3. This process works for any real, imaginary, fractional or decimal number. Matrix addition and subtraction can only be done on matrices with the same dimensions, by adding or subtracting the corresponding elements. For scalar multiplication, each element of the product matrix is the scalar multiplied by the corresponding element of the original matrix.

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2. 4.4A Multiplying a matrix by a number To multiply a matrix by a number, multiply every element by the number. For example:

3. Suppose we have the following matrix and we want to multiply it by 2.

4. Then (as with the adding and subtracting) we treat each element separately, multiplying each element by 2. So we get the following result:

5. Similarly to multiply the matrix by 3, we multiply each element by 3. Similarly for any number. So we can write the more general case of multiplying that matrix by any number c, say, as:

6. Now we can go one step further and make our rule still more general, by using our general matrix:

7. If we multiply this general matrix by any number c, we get this: So now we can multiply any matrix by any number. It's the same whether c is positive or negative, whole number, fraction or decimal, real or imaginary. It's also the same process whatever the shape of the matrix, whether it's 3x5 or 4x4 or any other combination.

8. Addition, Subtraction and Scalar Multiplication Matrices can be added and subtracted if they have the same dimensions (row x column). They can be altered by scalar multiplication. These operations can be done on the Casio calculator fairly easily and the process is helpful if you have lots to do with the same matrices. Let’s start with paper and pencil to add, subtract, and find scalar multiples for the following matrices.

9. Matrix Addition We can only add matrices of the same order . Example: Matrix addition is very simple; we just add the corresponding elements.

10. Matrix Subtraction Similarly, we can only subtract matrices of the same order. Example: We subtract the corresponding elements.

11. Scalar Multiplication of Matrices In matrix algebra, a real number is called a scalar . The scalar product of a real number, r , and a matrix A is the matrix rA. Each element of matrix rA is r times its corresponding element in A . Given scalar r and matrix

12. Example : Let A = , find 4 A . Solution:

13. Solving Matrix Equations A matrix equation is an equation in which a variable stands for a matrix . Example: Solve for the matrix X :

14. Solution