A system of linear equations can be written in a compact form using matrices. Instead of listing each equation separately, the coefficients go into a matrix, the variables go into a column vector, and the constants go into another column vector. This gives the matrix equation A x = b, which is easier to organize and solve.
Matrix form is especially useful when a system has many equations or many variables.
In A x = b, A is the coefficient matrix, x is the variable vector, and b is the constant vector. For the system 2x + y = 5 and x - y = 1, the matrix equation is [[2, 1], [1, -1]][[x], [y]] = [[5], [1]]. If A has an inverse, the solution is x = A^-1 b, where the bold x represents the variable vector.
This method connects algebraic systems to matrix operations used in science, engineering, economics, and computer graphics.
Key Facts
- Matrix form of a linear system is A x = b.
- A is the coefficient matrix, x is the variable vector, and b is the constant vector.
- For 2x + y = 5 and x - y = 1, A = [[2, 1], [1, -1]], x = [[x], [y]], and b = [[5], [1]].
- If A is invertible, the solution is x = A^-1 b.
- A matrix is invertible only if det(A) is not equal to 0.
- For a 2 by 2 matrix A = [[a, b], [c, d]], det(A) = ad - bc.
Vocabulary
- Linear system
- A set of linear equations that use the same variables and must be solved together.
- Coefficient matrix
- The matrix made from the numerical coefficients of the variables in a linear system.
- Variable vector
- A column matrix that lists the unknown variables in a fixed order.
- Constant vector
- A column matrix that lists the numbers on the right side of the equations.
- Inverse matrix
- A matrix A^-1 that reverses the effect of A when A^-1 A = I.
Common Mistakes to Avoid
- Mixing the order of variables, because the columns of the coefficient matrix must match the order in the variable vector.
- Putting constants into the coefficient matrix, because constants belong in the vector b, not in A.
- Using x = b A^-1, because matrix multiplication is order dependent and the correct inverse method is x = A^-1 b.
- Trying to use an inverse when det(A) = 0, because a matrix with determinant 0 has no inverse and the system cannot be solved by x = A^-1 b.
Practice Questions
- 1 Write the system 3x + 2y = 7 and x - 4y = -5 as a matrix equation A x = b.
- 2 For A = [[2, 1], [1, -1]] and b = [[5], [1]], find det(A), A^-1, and the solution vector x = A^-1 b.
- 3 Explain why the order of variables in the variable vector must match the order of columns in the coefficient matrix.