Construct inverses by elimination and recognize singularity.
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Try
The map A=[[1,2],[0,1]] sends (3,4) to (11,4). Recover the input by keeping the second coordinate and subtracting twice it from the first. The inverse matrix is [[1,−2],[0,1]]. Undoing a map is a structural property, not merely dividing each displayed entry.
Predict
Predict why a collapsed direction cannot be recovered. For a square matrix, an inverse B must satisfy AB=BA=I, where I leaves every vector unchanged. If A sends two distinct inputs to the same output, no single inverse can know which input to return.
Experiment
Augment A=[[1,1],[1,2]] with I and perform row operations on the full four-column array. Subtract row 1 from row 2, then subtract row 2 from row 1. The left block becomes I and the right becomes [[2,−1],[−1,1]]. Multiply the original by the result to check.
Understand
Worked reasoning: each elementary row operation is multiplication by an invertible elementary matrix. If their combined action E sends A to I, then EA=I; for a square matrix with a pivot in every column, this gives the inverse. Applying the same operations to I records E. If elimination cannot produce I on the left, A is singular.
For [[a,b],[c,d]], the inverse is (1/(ad−bc))[[d,−b],[−c,a]] when ad−bc≠0. This formula is a two-dimensional instance, not a substitute for general elimination. For square invertible A,B, the inverse of AB is B⁻¹A⁻¹: the last operation applied is the first undone.
Invalid step: cancel a matrix factor by division without establishing its inverse. A singular matrix can annihilate nonzero differences, so AB=AC need not imply B=C.
Apply
Check an alleged inverse by a product and preserve factor order. To solve one system, direct elimination is usually preferable to explicitly computing an inverse. The inverse still provides a powerful theorem connecting solvability, pivots and uniqueness.
Continue
Depth: rectangular matrices may have a left inverse or a right inverse without having a two-sided inverse. The square invertible-matrix equivalences must not be applied unchanged to rectangular maps. Numerical near-singularity is also distinct from exact singularity.
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Practice and reasoning
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For A=[[2,1],[0,3]], enter the (1,2) entry of A⁻¹.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A=[[1,2],[0,1]]. Solve Ax=(9,3). Enter x₁.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which matrix is singular?
For invertible square A and B of the same size, (AB)⁻¹ equals:
Find and verify the inverse of [[1,1],[1,2]].
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If B and C are both two-sided inverses of square A, prove B=C.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
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