Exam Response (matrices)

Exam Response (matrices)

What is it?

Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not. Write sizes before you multiply. One entry of a product per line if needed.

In simple words

• Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not.
• Write sizes before you multiply.
• One entry of a product per line if needed.
• Show the product check A A^-1 = I if marks allow.

Important terms

EXAM RESPONSE (MATRICES): Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not. Write sizes before you multiply. One entry of a product per line if needed.

1. COMPUTE: Add same size. Multiply row-col. det cofactors. Inverse / det. Rank = pivots.

2. SYSTEMS: det != 0 unique. 0=1 none. Free var infinite. Homogeneous always 0.

What you should learn

• State the main ideas and notation used in exam response (matrices).
• Apply the rules of exam response (matrices) in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

• ad-bc
• [A|I]
• AB != BA

COMPUTE

Add same size. Multiply row-col. det cofactors. Inverse / det. Rank = pivots.

SYSTEMS

det != 0 unique. 0=1 none. Free var infinite. Homogeneous always 0.

Full school-level explanation

Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not. Write sizes before you multiply. One entry of a product per line if needed.

Compute, then interpret:

WHAT — EXAM RESPONSE (MATRICES)

Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not. Write sizes before you multiply. One entry of a product per line if needed.

RULE — 1. COMPUTE

Add same size. Multiply row-col. det cofactors. Inverse / det. Rank = pivots.

METHOD — 2. SYSTEMS

det != 0 unique. 0=1 none. Free var infinite. Homogeneous always 0.

WORKED EXAMPLE

Worked 4-mark 2x2 det and inverse. A = [[4,3],[2,2]] det = 8-6 = 2 A^-1 = (12) [[2,-3],[-2,4]] = [[1, -32],[-1, 2]]

COMPUTE

Add same size. Multiply row-col. det cofactors. Inverse / det. Rank = pivots.

Remember

• size then arithmetic

Examples

• ad-bc
• [A|I]
• AB != BA

SYSTEMS

det != 0 unique. 0=1 none. Free var infinite. Homogeneous always 0.

Remember

• last row

Examples

• Cramer A_i
• Gauss back-sub
• Ax=0 trivial

How to answer a exam response (matrices) question

How it works

• Identify the given quantities, notation, and required result.
• Choose the matching rule and write it before substitution.
• Work one algebraic step at a time, keeping brackets and signs clear.
• Check restrictions, simplify the result, and verify it where possible.

Remember

• inner sizes match
• det=0: no unique inverse solution
• det 2
• check I
• size then arithmetic
• last row
• inner sizes match
• det=0: no unique inverse solution

How the ideas connect

Worked 4-mark 2x2 det and inverse. A = [[4,3],[2,2]] det = 8-6 = 2 A^-1 = (12) [[2,-3],[-2,4]] = [[1, -32],[-1, 2]]

Deeper understanding

Go beyond memorising Exam Response (matrices). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked 4-mark 2x2 det and inverse. A = [[4,3],[2,2]] det = 8-6 = 2 A^-1 = (12) [[2,-3],[-2,4]] = [[1, -32],[-1, 2]]

Common mistakes

Exam focus

Core Idea

1. COMPUTE

2. SYSTEMS

Worked Example

Exam focus

Quick revision

• inner sizes match
• det=0: no unique inverse solution
• det 2
• check I
• EXAM RESPONSE (MATRICES): Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consisten
• 1. COMPUTE: Add same size. Multiply row-col. det cofactors. Inverse / det. Rank = pivots.

Show the product check A A^-1 = I if marks allow.

For long answers on Exam Response (matrices): define clearly → explain the main parts → give one example → state one exam caution.