Many problems reduce to a system of linear equations, for example
2x+3yx−y=8=−1.
The inverse matrix method solves such a system by writing it as a single matrix equation and then undoing the coefficient matrix with its inverse — the matrix analogue of dividing.
Writing the system as AX=B
Collect the coefficients, the unknowns and the constants:
A=(213−1),X=(xy),B=(8−1),
so the whole system becomes AX=B.
The idea
For numbers, ax=b gives x=a−1b provided a=0. The same works for matrices: if A is invertible, multiply AX=B on the left by A−1:
A−1(AX)=A−1B⇒IX=A−1B⇒X=A−1B.
X=A−1B
Multiplying on the left matters — matrix products do not commute, so BA−1 would be wrong.
When it works
The inverse A−1 exists only when detA=0, so:
detA=0: the system is consistent with the unique solution X=A−1B.
detA=0: no inverse; the system is either inconsistent (no solution) or has infinitely many — handle it by another method.
Worked steps
For the system above, detA=(2)(−1)−(3)(1)=−5=0, and
Method: Proving an Implication via the Contrapositive (Matrix Case)
Use this whenever you're asked to "show that if [property A] then [property B]" about a matrix and the forward direction is awkward to argue directly, but the reverse (negated) direction follows immediately from a known formula.
Steps
Step 1: Write the statement as p⇒q
Identify p = the hypothesis (e.g. "A is invertible") and q = the conclusion (e.g. "A is non-singular, i.e. ∣A∣=0").
Step 2: Form and state the contrapositive ∼q⇒∼p
Negate and swap: "if A is singular (∣A∣=0), then A is not invertible." Confirm this is logically equivalent to the original — proving one proves the other.
Mistake 1: Arguing directly and circularly instead of via the contrapositive
Reasoning "since A is invertible, A−1=∣A∣adjA exists, so ∣A∣=0" skips straight to a forward algebraic argument without ever stating p⇒q and its contrapositive. Why it's wrong: the exercise specifically names the contrapositive technique — jumping to a bare forward statement misses the reasoning skill being tested and can leave the logical structure incompletely justified. Correct approach: explicitly name p and q, state the contrapositive ∼q⇒∼p, and prove that.
Mistake 2: Confusing "singular" and "non-singular" …
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
AHSEC Higher Secondary (HS) Final Examination 2020Set ANNUAL6 marks
Q.Using matrix method solve the following system of linear equations: x−y+z=4, 2x+y−3z=0, x+y+z=2.
OR
Using elementary transformation find the inverse of the following matrix: A=1−32305−2−50.
›Reveal solutionSolution
Solve AX=B via X=A−1B using the adjoint method; for the OR, row-reduce [A∣I] to [I∣A−1].
Matrix method: x−y+z=4,2x+y−3z=0,x+y+z=2
A=121−1111−31,X=xyz,B=402
detA=1(1+3)−(−1)(2+3)+1(2−1)=4+5+1=10=0, so a unique solution exists.
Since ∣A∣=−17e0, A−1 exists and X=A−1B gives a unique solution. Solving (by X=A−1B, or by elimination): from the second equation y=1−2x+z; substituting into the first gives 7x+z=10, and into the third gives 17x=17.