Matrices
Matrix inverse worksheets
Decide whether six numeric 2 × 2 matrices have inverses, then find six exact inverses or state that no inverse exists.
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Matrix equations worksheets
Solve six exact matrix equations by isolating a scalar multiple of X or multiplying by a 2 × 2 inverse on the left.
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For A + cX = B, subtract A and divide every entry by the nonzero scalar c. For AX = B, find A⁻¹ and multiply on the left: X = A⁻¹B. Keep fractions exact and check the original equation.
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Customize this worksheetApply each inverse operation to the whole matrix
For a direct equation A + cX = B, begin with cX = B − A. The matrices must have matching dimensions, so subtraction happens entry by entry. Then multiply every entry of B − A by 1/c. If c is negative, that sign belongs in the reciprocal and affects every entry of X.
For example, if A has rows (1, −2) and (3, 0), c = −2, and B has rows (−5, 4) and (1, −6), then B − A has rows (−6, 6) and (−2, −6). Dividing by −2 gives X with rows (3, −3) and (1, 3). Recombining A − 2X returns B.
Direct mode includes rectangular as well as square matrices. A matrix does not need to be square when the only remaining operation is division by a nonzero scalar.
Learning objective
What this activity builds
Preserve matrix order while solving A + cX = B or AX = B, and check the result by recomposing the original equation.
Worked example
Solve AX = B when A has rows (1, 2) and (0, 1), and B has rows (5) and (2).
X has rows (1) and (2).
A⁻¹ has rows (1, −2) and (0, 1), so X = A⁻¹B = (1, 2)ᵀ. The inverse belongs on the left: BA⁻¹ is the mistaken right-side product, and it is not even defined for this 2 × 1 B.
Teaching notes
Make the practice count
- Name the operation applied to the whole matrix equation before calculating entries.
- Treat a negative coefficient as a signed scalar; dividing by it changes every entry’s sign.
- For AX = B, write A⁻¹AX = A⁻¹B so the cancellation and multiplication order stay visible.
- Substitute the completed matrix into the original equation to catch arithmetic and order errors.
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Choose equations that isolate a signed scalar multiple of X or equations that require a 2 × 2 inverse. Each worksheet has six exact problems, and its separate key preserves matrix multiplication order.
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