LEARN · EXPLAIN · REVISE

Show AB≠BA for the specified matrices.

Read the idea, work independently, then explain what changed.

2027 JM02 考試大綱 · 3. Systems, matrices and determinants · PDF 2 / printed page 2

TOPIC 01

Matrices, determinants and linear systems

Add and multiply matrices, use determinants and solve systems without assuming invertibility.

Add and multiply matrices, use determinants and solve systems without assuming invertibility.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

A=\begin{pmatrix}1&12\\0&1\end{pmatrix},\ B=\begin{pmatrix}1&0\\0&0\end{pmatrix}

Matrix products require matched inner dimensions; a square matrix has an inverse only if its determinant is nonzero.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Compute both products separately.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$(AB)_{12}=0,\quad(BA)_{12}=12\ne0$
One unequal entry is sufficient to disprove commutativity.

The requested relation or conclusion is shown below.

AB=\begin{pmatrix}1&0\\0&0\end{pmatrix},\ BA=\begin{pmatrix}1&12\\0&0\end{pmatrix}

Checks and common pitfalls: One unequal entry is sufficient to disprove commutativity.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Add and multiply matrices, use determinants and solve systems without assuming invertibility.
Which condition is essential in matrices, determinants and linear systems?
Can AB and BA have different entries even when both products exist?

Model or definition: Add and multiply matrices, use determinants and solve systems without assuming invertibility.
$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$
Conditions: Matrix products require matched inner dimensions; a square matrix has an inverse only if its determinant is nonzero.