Verify an Inverse by Producing the Identity

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Which matrix is the inverse of [[1,2],[0,1]]? Candidate answers: (A) [[0,1],[1,2]] (B) [[1,0],[−2,1]] (C) [[1,2],[0,1]] (D) [[1,−2],[0,1]].
• Independent check inputs: Which matrix is the inverse of [[1,3],[0,1]]? Candidate answers: (A) [[0,1],[1,3]] (B) [[1,0],[−3,1]] (C) [[1,3],[0,1]] (D) [[1,−3],[0,1]].

Learn the idea

Matrices organize linear operations and systems through rows and columns. Multiplication uses row-column products, a determinant tests two-dimensional invertibility, and valid row operations preserve a system’s solution set when applied to every entry. An inverse matrix undoes a matrix’s linear operation. Multiplying the proposed pair should give ones on the main diagonal and zeros elsewhere, which defines the identity matrix.

Here is the method on the supplied teaching case. Multiply [[1,2],[0,1]] by [[1,−2],[0,1]]. The off-diagonal sum −2+2=0 and diagonal entries remain 1, giving the identity matrix.

The inverse is not formed by reciprocating every entry. Check a product in the correct row-column order; singular matrices have no inverse, so existence must not be assumed for every matrix.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Which matrix is the inverse of [[1,9],[0,1]]? Candidate answers: (A) [[0,1],[1,9]] (B) [[1,0],[−9,1]] (C) [[1,9],[0,1]] (D) [[1,−9],[0,1]]. Show the mathematical reason, not only the selected answer. Expected result: [[1,−9],[0,1]]. Multiply [[1,9],[0,1]] by [[1,−9],[0,1]]. The off-diagonal sum −9+9=0 and diagonal entries remain 1, giving the identity matrix.

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