The determinant is a number; the inverse, when it exists, is a matrix. Keep those objects separate. A quick determinant calculation can tell you whether an inverse method is even available before you spend time solving a system.
Curriculum checked 2 October 2026 · original practice for selected skills.
You will need: Matrix multiplication, simultaneous equations and fraction arithmetic.
Attempt each task before opening a hint. If you need help, reveal the first hint, then the setup, then compare your written steps with the solution. Finish with the retests without referring back.
For a 2 × 2 matrix with rows (a,b) and (c,d), the determinant is ad − bc. The second product is subtracted as a whole.
A zero determinant means the square matrix is singular and has no inverse. It does not, by itself, distinguish an inconsistent system from one with infinitely many solutions.
For a non-zero determinant, swap a and d, negate b and c, then divide the resulting matrix by ad − bc. Check by multiplying with the original matrix.
Write both diagonal products before subtracting.
The second product is (−2)(3) = −6.
Subtraction of a negative becomes addition.
The non-zero determinant permits an inverse.
Answer: 10
Before moving on: can you explain why your method works, as well as give the answer?
Write a method as well as an answer. The tasks change the reasoning, not just the numbers.
Compute the determinant first.
Its determinant is 2 − 1 = 1.
Swap the diagonal entries and negate the other two.
Dividing by 1 leaves the adjugate. Multiplying by A gives the identity.
Answer:
Before moving on: can you explain why your method works, as well as give the answer?
Singular means determinant zero.
Solve 6k − 6 = 0.
Use both cross products.
Divide by 6 to obtain the only value.
Answer: k = 1
Before moving on: can you explain why your method works, as well as give the answer?
Subtract the second equation from the first.
The coefficient determinant is 2 × 1 − 1 × 1.
Subtraction gives x = 3; use the second equation for y = 1.
The determinant is 1, so the matrix is invertible. Both original equations hold.
Answer: x = 3, y = 1; the inverse exists.
Before moving on: can you explain why your method works, as well as give the answer?
A tempting claim: “A zero determinant means every corresponding system has no solution.”
The constants also matter: proportional equations may describe the same line or two parallel lines.
These changed questions test whether you can reconstruct the method. If you use a hint, record where you got stuck and retry later. Completing this small set does not establish full chapter mastery.
Retain the negative entry in brackets.
Use 3(4) − (−1)(2).
The products are 12 and −2.
It is non-zero, so this matrix is invertible.
Answer: 14
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
Eliminate y by subtraction.
Two x equals 6.
x = 3 and y = 2.
3(3)+2=11 and 3+2=5.
Answer: x = 3, y = 2
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
If a retest exposed the same error, rewrite the first incorrect step and explain its correction aloud. If both were independent, return to a mixed exercise or a missed paper question. A parent can ask what changed in the method rather than only asking for the answer.
No. This set concentrates on 2 × 2 reasoning. The Class 12 syllabus also includes determinants up to order three, cofactors, triangle area and other system questions.
Yes. Use the two print buttons above. The question sheet leaves working space and hides solutions; the worked-solutions option includes the solution steps. Your browser can save either view as a PDF.
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