Matrices · Form 5
What are the determinant and inverse of a matrix for?
The determinant is a single number that tells you whether a matrix has an inverse; the inverse is the matrix that undoes it, mainly used to solve simultaneous linear equations.
The determinant: a yes-or-no test
For a 2 × 2 matrix with rows (a, b) and (c, d), the determinant is the single number ad − bc. Its job is to test the matrix: if the determinant is not zero, the matrix has an inverse; if it is zero, no inverse exists.
Working it out first tells you at once whether the rest of the problem is even possible.
The inverse: the matrix that undoes another
The inverse of a matrix A, written A⁻¹, is the matrix that gives the identity matrix when you multiply the two together, the identity being the matrix version of the number 1. Its main use in SPM is to solve a pair of simultaneous linear equations written in matrix form: multiplying by the inverse isolates the unknowns in one clean move.
Think of it as the matrix way of doing what dividing does for ordinary numbers.
You multiply by the inverse, you never divide
There is no such thing as dividing by a matrix, and the inverse is not simply one over the matrix. Instead, wherever you would divide with numbers, you multiply by the inverse with matrices.
And a matrix whose determinant is zero has no inverse at all, the equations it stands for then have no unique solution, so checking the determinant first saves you from a dead end.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What are the determinant and inverse of a matrix actually for?
The determinant of a 2×2 matrix checks whether an inverse exists, if it equals 0, that matrix has no inverse. When an inverse does exist, it lets you 'undo' a matrix, which is mainly used to solve a pair of simultaneous linear equations by writing them in matrix form.
How are determinant and inverse of a matrix tested in SPM?
Paper 1 usually asks you to find the determinant or inverse of a given 2×2 matrix directly, testing whether you remember the formula. Paper 2 typically goes further, asking you to write simultaneous equations as a matrix equation, find the inverse, then multiply to find the values of the unknowns.
What is a common mistake when finding the inverse of a matrix?
A frequent mistake is calculating the determinant as ad + bc instead of ad - bc, which gives a wrong inverse for every step that follows. Students also forget that when the determinant equals 0, the matrix has no inverse at all, so that method cannot be used to solve the equations.