MathematicsChapter 2: Matrices and Determinants
Inverse and Rank (row Operations)
A square matrix is invertible if there is B with AB=BA=I.
Learning objectives
- State the main ideas and notation used in inverse and rank (row operations).
- Apply the rules of inverse and rank (row operations) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
A square matrix is invertible if there is B with AB=BA=I. Then B = A^{-1}. For 2x2, A^{-1} = (1/det A) [[d,-b],[-c,a]] if det != 0. For 3x3, use Gauss-Jordan: put [A | I] and row-reduce to [I | A^{-1}]. Rank is the number of non-zero rows after row-echelon (number of pivot positions). Meaning: rank counts independent rows/columns. Full rank n for n x n means invertible.
Mathematical form
det = 6-5 = 1
AB=BA
B = A
det = 6-5
Where / symbols
- INVERSE AND RANK (ROW OPERATIONS)
- A square matrix is invertible if there is B with AB=BA=I. Then B = A^{-1}. For 2x2, A^{-1} = (1/det A) [[d,-b],[-c,a]] if det != 0. For 3x3, use Gauss-Jordan: put [A | I] and row-reduce to [I | A^{-1}]. Rank is the number of non-zero rows after row-echelon (number of pivot positions). Meaning: rank counts independent rows/columns. Full rank n for n x n means invertible.
- 1. FIND A^{-1}
- If det=0, stop: no inverse. 2x2 swap formula. 3x3: row ops on the augmented identity. Check A A^{-1} = I on one row if time.
- 2. RANK
- Swap, scale, add rows (elementary ops). Row-echelon: staircase of pivots. Rank 0 is the zero matrix. Rank < n: singular if square.
Relationship
- invertible <-> det != 0 (square)
- rank = number of pivots
Overview
Inverse, then rank:
Find A^{-1}
If det=0, stop: no inverse. 2x2 swap formula. 3x3: row ops on the augmented identity. Check A A-1 = I on one row if time.
Examples 2x2 adjugate / det · [A|I] -> [I|A^{-1}] · No inverse if a row becomes 0 | junk
Rank
Swap, scale, add rows (elementary ops). Row-echelon: staircase of pivots. Rank 0 is the zero matrix. Rank < n: singular if square.
Examples Rank 2 for a typical invertible 2x2 · Two equal rows: rank drops · Complex entries: same ops
WHAT — INVERSE AND RANK (ROW OPERATIONS)
A square matrix is invertible if there is B with AB=BA=I. Then B = A^{-1}. For 2x2, A^{-1} = (1/det A) [[d,-b],[-c,a]] if det != 0. For 3x3, use Gauss-Jordan: put [A | I] and row-reduce to [I | A^{-1}]. Rank is the number of non-zero rows after row-echelon (number of pivot positions). Meaning: rank counts independent rows/columns. Full rank n for n x n means invertible.
RULE — 1. FIND A^{-1}
If det=0, stop: no inverse. 2x2 swap formula. 3x3: row ops on the augmented identity. Check A A-1 = I on one row if time.
METHOD — 2. RANK
Swap, scale, add rows (elementary ops). Row-echelon: staircase of pivots. Rank 0 is the zero matrix. Rank < n: singular if square.
WORKED EXAMPLE
Worked 2x2 inverse of [[2,1],[5,3]]. det = 6-5 = 1 A-1 = [[3,-1],[-5,2]] check: first row of product [2,1][[3,-1],[-5,2]] = [1,0]
How to answer a inverse and rank (row operations) question
- 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.
Worked example
- Worked 2x2 inverse of [[2,1],[5,3]].
- det = 6-5 = 1
- A-1 = [[3,-1],[-5,2]]
- check: first row of product [2,1][[3,-1],[-5,2]] = [1,0]
In simple words
If det=0 there is no inverse. Do not divide by zero.
Key points
- invertible <-> det != 0 (square)
- rank = number of pivots
- det first
- pivots
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Inverse and Rank (row Operations)
Sticky Notes · Quick Revision
Core Idea
- A square matrix is invertible if there is B with AB=BA=I.
- Then B = A-1.
- For 2x2, A-1 = (1/det A) [[d,-b],[-c,a]] if det != 0.
1. FIND A^{-1}
- If det=0, stop: no inverse.
- 2x2 swap formula.
- 3x3: row ops on the augmented identity.
- Check A A-1 = I on one row if time.
2. RANK
- Swap, scale, add rows (elementary ops).
- Row-echelon: staircase of pivots.
- Rank 0 is the zero matrix.
- Rank < n: singular if square.
Worked Example
- Worked 2x2 inverse of [[2,1],[5,3]].
- det = 6-5 = 1
- A-1 = [[3,-1],[-5,2]]
- check: first row of product [2,1][[3,-1],[-5,2]] = [1,0]
Common Misconception
- If det=0 there is no inverse. Do not divide by zero.
FBISE Exam Check
- State the rule for inverse and rank (row operations) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- A square matrix is invertible if there is B with AB=BA=I.
- Then B = A-1.
- For 2x2, A-1 = (1/det A) [[d,-b],[-c,a]] if det != 0.
- If det=0 there is no inverse. Do not divide by zero.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.