Solving Systems with Matrices
Linear algebra provides powerful tools for solving systems of linear equations, especially when there are many variables.
Representing Systems with Matrices
A system of linear equations like:
2x + 4y = 10
3x - y = 5
Can be written in matrix form Ax = b:
[ 2 4 ] [ x ] = [ 10 ]
[ 3 -1 ] [ y ] [ 5 ]
A is the coefficient matrix.
x is the variable vector.
b is the constant vector.
Gaussian Elimination
This is a systematic algorithm for solving a system of linear equations by transforming its augmented matrix into row-echelon form.
Augmented Matrix: The coefficient matrix with the constant vector appended as a column.
[ 2 4 | 10 ]
[ 3 -1 | 5 ]
Allowed Row Operations:
1.Swap two rows.
2.Multiply a row by a non-zero constant.
3.Add a multiple of one row to another row.
Goal (Row-Echelon Form): Create an upper triangular matrix, where the first non-zero entry in each row is 1, and all entries below the main diagonal are zero.
[ 1 | ]
[ 0 1 | * ]
Once in this form, the solution can be found easily through back-substitution.
The Identity and Inverse Matrix
Identity Matrix (I): A square matrix with ones on the main diagonal and zeros everywhere else. It is the matrix equivalent of the number '1'. Multiplying any matrix by the identity matrix leaves it unchanged (AI = A).
[ 1 0 ]
[ 0 1 ] (This is the 2x2 identity matrix)
Inverse Matrix (A⁻¹): For a square matrix A, its inverse A⁻¹ is a matrix such that AA⁻¹ = A⁻¹A = I. Not all matrices have an inverse. A matrix has an inverse if and only if its determinant is non-zero.
Solving Ax = b with an Inverse: If the inverse exists, we can solve for the variables by multiplying both sides by A⁻¹:
A⁻¹(Ax) = A⁻¹b
(A⁻¹A)x = A⁻¹b
Ix = A⁻¹b
x = A⁻¹b