INVERSE OF A 2x2 MATRIX

INVERSE OF A 2x2 MATRIX

What is it?

If |A| != 0, A^-1 = (1/|A|) adj(A). For [[a,b],[c,d]], adj(A) = [[d, -b], [-c, a]] (swap the main diagonal, negate the other two). Then A A^-1 = I. If |A|=0, stop: no inverse.

In simple words

• If |A| != 0, A^-1 = (1/|A|) adj(A).
• ADJOINT: A=[[2,3],[1,4]] has adj [[4,-3],[-1,2]].
• SCALE BY 1/|A|: |A|=8-3=5, so A^-1=(15)[[4,-3],[-1,2]] = [[45, -35], [-15, 25]].
• Worked-method reminder: A^-1 = [[45, -35], [-15, 25]]

Important terms

INVERSE OF A 2x2 MATRIX: If |A| != 0, A^{-1} = (1/|A|) adj(A). For [[a,b],[c,d]], adj(A) = [[d, -b], [-c, a]] (swap the main diagonal, negate the other two). Then A A^{-1} = I. If |A|=0, stop: no inverse.

ADJOINT: A=[[2,3],[1,4]] has adj [[4,-3],[-1,2]]. Swap 2 with 4, flip signs of 3 and 1. Do not swap a with d and also swap b with c wrongly.

SCALE BY 1/|A|: |A|=8-3=5, so A^-1=(15)[[4,-3],[-1,2]] = [[45, -35], [-15, 25]]. Check: A times this should be I.

What you should learn

• Explain inverse of a 2x2 matrix using the correct definitions, conditions, and notation.
• Apply (1/|A|) adj(A), swap diagonal, negate off correctly in routine and exam-style questions.
• Show each important step, simplify the result, and check that it satisfies the original conditions.

Examples

• adj[[2,3],[1,4]]=[[4,-3],[-1,2]]
• Not the transpose only
• Not without the minuses

ADJOINT

A=[[2,3],[1,4]] has adj [[4,-3],[-1,2]]. Swap 2 with 4, flip signs of 3 and 1. Do not swap a with d and also swap b with c wrongly.

SCALE BY 1/|A|

|A|=8-3=5, so A^-1=(15)[[4,-3],[-1,2]] = [[45, -35], [-15, 25]]. Check: A times this should be I.

Full school-level explanation

If |A| != 0, A^-1 = (1/|A|) adj(A). For [[a,b],[c,d]], adj(A) = [[d, -b], [-c, a]] (swap the main diagonal, negate the other two). Then A A^-1 = I. If |A|=0, stop: no inverse.

Det first, then adjoint:

WHAT — INVERSE OF A 2x2 MATRIX

If |A| != 0, A^{-1} = (1/|A|) adj(A). For [[a,b],[c,d]], adj(A) = [[d, -b], [-c, a]] (swap the main diagonal, negate the other two). Then A A^{-1} = I. If |A|=0, stop: no inverse. The symbols and conditions are part of the mathematics, so include them when stating the rule.

WHY — Why this topic matters

Inverse Of A 2X2 Matrix matters because matrices organise numbers and give short methods for transformations and simultaneous equations. Correct order, determinant, and multiplication rules prevent most exam errors.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through ADJOINT, then SCALE BY 1/|A|. Worked: A = [[2, 3], [1, 4]]. Find A^-1. |A| = 24 - 31 = 8 - 3 = 5 adj(A) = [[4, -3], [-1, 2]] A^-1 = (15) [[4, -3], [-1, 2]] A^-1 = [[45, -35], [-15, 25]] Finish by checking signs, restrictions, units, and whether the answer matches the question.

ADJOINT

A=[[2,3],[1,4]] has adj [[4,-3],[-1,2]]. Swap 2 with 4, flip signs of 3 and 1. Do not swap a with d and also swap b with c wrongly.

Remember

• [[d,-b],[-c,a]]

Examples

• adj[[2,3],[1,4]]=[[4,-3],[-1,2]]
• Not the transpose only
• Not without the minuses

SCALE BY 1/|A|

|A|=8-3=5, so A^-1=(15)[[4,-3],[-1,2]] = [[45, -35], [-15, 25]]. Check: A times this should be I.

Remember

• divide every adj entry

Examples

• |A|=5
• A^-1 exists
• AA^-1=I

Remember

• (1/|A|) adj(A)
• swap diagonal, negate off
• [[45, -35], [-15, 25]]
• no inverse if det 0
• [[d,-b],[-c,a]]
• divide every adj entry
• (1/|A|) adj(A)
• swap diagonal, negate off

How the ideas connect

Worked: A = [[2, 3], [1, 4]]. Find A^-1. |A| = 24 - 31 = 8 - 3 = 5 adj(A) = [[4, -3], [-1, 2]] A^-1 = (15) [[4, -3], [-1, 2]] A^-1 = [[45, -35], [-15, 25]]

Deeper understanding

Go beyond memorising INVERSE OF A 2x2 MATRIX. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: A = [[2, 3], [1, 4]]. Find A^-1. |A| = 24 - 31 = 8 - 3 = 5 adj(A) = [[4, -3], [-1, 2]] A^-1 = (15) [[4, -3], [-1, 2]] A^-1 = [[45, -35], [-15, 25]]

Common mistakes

Exam focus

Core Idea

ADJOINT

SCALE BY 1/|A|

Worked Method

Exam focus

Quick revision

• (1/|A|) adj(A)
• swap diagonal, negate off
• [[45, -35], [-15, 25]]
• no inverse if det 0
• INVERSE OF A 2x2 MATRIX: If |A| != 0, A^-1 = (1/|A|) adj(A). For [[a,b],[c,d]], adj(A) = [[d, -b], [-c, a]] (swap the main
• ADJOINT: A=[[2,3],[1,4]] has adj [[4,-3],[-1,2]]. Swap 2 with 4, flip signs of 3 and 1. Do not swap a with d

If |A|=0 you cannot 'divide by zero' to make an inverse. Write 'does not exist'.

For long answers on INVERSE OF A 2x2 MATRIX: define clearly → explain the main parts → give one example → state one exam caution.