Matrix addition and subtraction may only be performed on matrices of the same order. Matrix addition is simple addition of the entries in one matrix to the corresponding entries in the other matrix. Matrix subtraction is simple subtraction of entries in one matrix from the corresponding entries in the other matrix. A square matrix has an equal number of row and columns. Two matrices are equal if their corresponding entries are equal and they are both the same order.
Scalar multiplication of a matrix is simple multiplication of each entry in a matrix by a constant. Two matrices can only be multiplied if they have the same number of entries. Columns can be multiplied times rows. A row-matrix has only one row, and a column-matrix has only one column. If a row-matrix and a column-matrix have the same number of entries, the row-column product is the products of their corresponding entries added together. The row-matrix must be on the left, and the column-matrix must be on the right. If the row-matrix has a different number of entries than the column-matrix, the matrices can not be multiplied.
Matrices are used to solve systems of linear equations. A system of three linear equations:
A1 X + B1 Y + C1 Z = D1
A2 X + B2 Y + C2 Z = D2
A3 X + B3 Y + C3 Z = D3
The coefficients and constants form an augmented matrix of the following form:
| A1 B1 C1 | D1 |
| A2 B2 C2 | D2 |
| A3 B3 C3 | D3 |
The Gauss Jordan method of solving the system of linear equations is the elimination method. The Gauss Jordan method transforms the matrix into the identity matrix. Multiplying matrix rows by non-zero numbers, adding common multiples of one row to entries in another row and interchanging rows, eliminates coefficients to solve the equations. The result, in this case, is X = A, Y = B and Z = C:
| 1 0 0 | A |
| 0 1 0 | B |
| 0 0 1 | C |
Matrices A and A1 are inverses if their product is the identity matrix. To determine the inverse of a matrix, create the augmented matrix formed by the coefficients and constants of the system of linear equations. Use row operations to obtain a 1 in row 1 column 1 and 0 in all other column 1 positions. Obtain a 1 in row 2 column 2 and zero in all other column 2 positions. Continue until you have created an identity matrix. The right half of the equation is the inverse matrix. Some matrices do not have inverses. A system of linear equations represented by AX = B can be solved by finding the inverse of A and multiplying matrix A by its inverse.