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Matrices, determinants and linear systems

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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.

What you will be able to explain

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

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.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Can AB and BA have different entries even when both products exist?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

A=[1,a;0,1], B=[1,0;b,1]. AB=[3,1;2,1], BA=[1,1;2,3]. The plotted columns belong to M=[1,a;b,1], with determinant -1 and area |det M|.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Compare a shear and a coordinate projection in both orders.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find det A.

A=\begin{pmatrix}2&1\\2&3\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

Use this intermediate relation.

det A=ad-bc

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$det A=3(2)-2=4$
Multiply diagonals with the correct subtraction order.

The requested value is 4.

Checks and common pitfalls: Multiply diagonals with the correct subtraction order.

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

02 / Standard#Worked example

Find the determinant of the triangular 3×3 matrix.

A=\begin{pmatrix}3&1&2\\0&2&3\\0&0&4\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

A triangular determinant is the diagonal product.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$det A=3\cdot2\cdot4=24$
Off-diagonal entries do not change a triangular determinant.

The requested value is 24.

Checks and common pitfalls: Off-diagonal entries do not change a triangular determinant.

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

03 / Transfer#Worked example

Show AB≠BA for the specified matrices.

A=\begin{pmatrix}1&4\\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.
$(AB)_{12}=0,\quad(BA)_{12}=4\ne0$
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&4\\0&0\end{pmatrix}

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

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

04 / Foundation#Your turn

Find det A.

A=\begin{pmatrix}5&1\\2&3\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

Use this intermediate relation.

det A=ad-bc

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$det A=3(5)-2=13$
Multiply diagonals with the correct subtraction order.

The requested value is 13.

Checks and common pitfalls: Multiply diagonals with the correct subtraction order.

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

05 / Foundation#Your turn

Find the (1,2) entry of AB.

A=\begin{pmatrix}1&6\\0&1\end{pmatrix},\ B=\begin{pmatrix}2&3\\1&4\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

Use this intermediate relation.

(AB)_{12}=A_{11}B_{12}+A_{12}B_{22}

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$(AB)_{12}=3+4(6)=27$
Use a row of A and a column of B, not matching positions.

The requested value is 27.

Checks and common pitfalls: Use a row of A and a column of B, not matching positions.

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

06 / Foundation#Your turn

Find x in the system.

x+y=9,\quad x-y=7

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

Add equations to eliminate y.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$2x=16\Rightarrow x=8$
The coefficient determinant is nonzero, so the solution is unique.

The requested value is 8.

Checks and common pitfalls: The coefficient determinant is nonzero, so the solution is unique.

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

07 / Foundation#Your turn

Find A⁻¹.

A=\begin{pmatrix}1&8\\0&1\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

Use the triangular inverse and verify AA^{-1}=I.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$\begin{pmatrix}1&8\\0&1\end{pmatrix}\begin{pmatrix}1&-8\\0&1\end{pmatrix}=I$
An explicit product verifies the inverse without relying on a memorized sign pattern.

The requested relation or conclusion is shown below.

A^{-1}=\begin{pmatrix}1&-8\\0&1\end{pmatrix}

Checks and common pitfalls: An explicit product verifies the inverse without relying on a memorized sign pattern.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

08 / Standard#Your turn

Find A⁻¹.

A=\begin{pmatrix}1&9\\0&1\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

Use the triangular inverse and verify AA^{-1}=I.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$\begin{pmatrix}1&9\\0&1\end{pmatrix}\begin{pmatrix}1&-9\\0&1\end{pmatrix}=I$
An explicit product verifies the inverse without relying on a memorized sign pattern.

The requested relation or conclusion is shown below.

A^{-1}=\begin{pmatrix}1&-9\\0&1\end{pmatrix}

Checks and common pitfalls: An explicit product verifies the inverse without relying on a memorized sign pattern.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

09 / Standard#Your turn

Find the determinant of the triangular 3×3 matrix.

A=\begin{pmatrix}10&1&2\\0&2&3\\0&0&4\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

A triangular determinant is the diagonal product.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$det A=10\cdot2\cdot4=80$
Off-diagonal entries do not change a triangular determinant.

The requested value is 80.

Checks and common pitfalls: Off-diagonal entries do not change a triangular determinant.

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

10 / Standard#Your turn

Classify the solutions of the system.

x+y=11,\quad2x+2y=22

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

The second equation is twice the first.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$y=11-x$
A zero determinant does not by itself mean no solution; this consistent system has infinitely many.

The requested relation or conclusion is shown below.

(x,y)=(s,11-s),\ s\in\mathbb R

Checks and common pitfalls: A zero determinant does not by itself mean no solution; this consistent system has infinitely many.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

11 / Standard#Your turn

Show AB≠BA for the specified matrices.

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

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

12 / Transfer#Your turn

Classify the solutions of the system.

x+y=13,\quad2x+2y=26

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

The second equation is twice the first.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$y=13-x$
A zero determinant does not by itself mean no solution; this consistent system has infinitely many.

The requested relation or conclusion is shown below.

(x,y)=(s,13-s),\ s\in\mathbb R

Checks and common pitfalls: A zero determinant does not by itself mean no solution; this consistent system has infinitely many.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

13 / Transfer#Your turn

Show AB≠BA for the specified matrices.

A=\begin{pmatrix}1&14\\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.
$(AB)_{12}=0,\quad(BA)_{12}=14\ne0$
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&14\\0&0\end{pmatrix}

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

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

End-of-lesson check and correction

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Teacher preparation and assessment

Question sequence

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?

Board plan

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.

Anticipated thinking

Matrix multiplication is not generally commutative.

Assessment checklist

1 mark: choose the correct representation and conditions.
1 mark: establish the intermediate relation.
1 mark: complete a connected calculation or proof.
1 mark: interpret and check the conclusion.