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Matrices
Matrices are powerful rectangular arrays of numbers that help us organise information and solve complex problems efficiently. They act as a shorthand for tasks like solving simultaneous equations or describing geometric transformations.
Need to know
What you need to know
- A system of equations $\mathbf{A}\mathbf{x} = \mathbf{b}$ has a unique solution if and only if $\det(\mathbf{A}) \neq 0$.
- The unique solution is given by $\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}$.
- If $\det(\mathbf{A}) = 0$, the system has either no solutions or infinitely many solutions. The matrix inverse method cannot be used.
Explanation
Matrices: The Ultimate Organiser
- Calculate the determinant of the 3x3 matrix. If it's zero, stop; the inverse does not exist.
- Find the matrix of minors. Then, create the matrix of cofactors by applying the 'checkerboard' pattern of signs (+, -, +, -,...).
- Transpose the matrix of cofactors to get the adjugate (or adjoint) matrix.
- The inverse is found by multiplying the adjugate matrix by 1 divided by the determinant.