medium difficulty8 min

Minors and Cofactors

The minor Mij of an element aij is the determinant obtained by deleting the ith row and jth column. The cofactor Aij is defined as Aij = (-1)^i+j Mij.

In one line: Minor means make it smaller by deleting; cofactor means attach the checkerboard sign.

Practice This Concept

Understand it

A minor is the determinant of the smaller matrix that remains after fixing the position of an element and deleting its row and column.
A cofactor is not just this minor; it is the signed minor, where the sign comes from (-1)^i+j.
This sign pattern is what makes cofactor expansion work for determinants of order 3 and above.
The same cofactors are later arranged into a cofactor matrix and transposed to form adj(A), so accuracy here directly affects inverse questions.

Example

For A = [[1,2,3],[4,5,6],[7,8,9]], M12 is found by deleting row 1 and column 2: M12 = |[[4,6],[7,9]]| = 36 - 42 = -6. A12 = (-1)³ M12 = 6.

Formula and rule

Mij = determinant left after deleting row i and column j.

Aij = (-1)^i+jMij.

Expansion along row i: |A| = ai1Ai1 + ai2Ai2 + ai3Ai3 for a 3 by 3 matrix.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

For A = [[2,1,0],[3,4,5],[1,0,6]], find A13.

1Delete row 1 and column 3 to get [[3,4],[1,0]].
2M13 = 3×0 - 4×1 = -4.
3Since (-1)¹⁺³ = 1, A13 = -4.

Method you can reuse

1Identify the position (i,j).
2Delete row i and column j.
3Find the remaining determinant.
4Apply the sign (-1)^i+j.
5Use the cofactor in expansion if required.

Visual tip

No separate diagram is required. A highlighted matrix grid can show the deleted row and column when introducing Mij.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If M23 = 5, what is cofactor A23?

Think first, then show answer

A23 = (-1)²⁺³ × 5 = -5.

Common wrong answer

Using Mij as the cofactor without applying the sign factor.

Don't mix this up

Deleting the wrong row or column while finding Mij. For M23, students must delete row 2 and column 3; deleting row 3 and column 2 gives a different minor and usually leads to a wrong cofactor sign as well.

Exam tip

Write the sign pattern [[+,-,+],[-,+,-],[+,-,+]] before calculating cofactors of a 3 by 3 matrix.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may ask for a specific minor, a specific cofactor, or determinant evaluation using cofactor expansion.

1
Enough for 1 mark
The minor Mij of an element aij is the determinant obtained by deleting the ith row and jth column. The cofactor Aij is defined as Aij = (-1)^i+j Mij.
2
Add for 2 marks
Example: For A = [[1,2,3],[4,5,6],[7,8,9]], M12 is found by deleting row 1 and column 2: M12 = |[[4,6],[7,9]]| = 36 - 42 = -6. A12 = (-1)³ M12 = 6.
3
Add for 3 marks
A cofactor is not just this minor; it is the signed minor, where the sign comes from (-1)^i+j.

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