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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

    Using Mij as the cofactor without applying the sign factor.

Definition

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.

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.

Rule to remember

Mij = determinant left after deleting row i and column j. Aij = (-1)^(i+j)Mij. Expansion along row i: |A| = ai1Ai1 + ai2Ai2 + ai3Ai3 for a 3 by 3 matrix.

Memory hook

Minor means make it smaller by deleting; cofactor means attach the checkerboard sign.

Examples and method

Worked example

For A = [[2,1,0],[3,4,5],[1,0,6]], find A13. Delete row 1 and column 3 to get [[3,4],[1,0]]. M13 = 3×0 - 4×1 = -4. Since (-1)^(1+3) = 1, A13 = -4.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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.

Common wrong answer

Using Mij as the cofactor without applying the sign factor.

Exam tip

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

Quick check

If M23 = 5, what is cofactor A23?

A23 = (-1)^(2+3) × 5 = -5.

Answer writing and exam use

1-mark answer

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-mark answer

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. Mij = determinant left after deleting row i and column j. Aij = (-1)^(i+j)Mij. Expansion along row i: |A| = ai1Ai1 + ai2Ai2 + ai3Ai3 for a 3 by 3 matrix. 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-mark answer

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. Mij = determinant left after deleting row i and column j. Aij = (-1)^(i+j)Mij. Expansion along row i: |A| = ai1Ai1 + ai2Ai2 + ai3Ai3 for a 3 by 3 matrix. For A = [[2,1,0],[3,4,5],[1,0,6]], find A13. Delete row 1 and column 3 to get [[3,4],[1,0]]. M13 = 3×0 - 4×1 = -4. Since (-1)^(1+3) = 1, A13 = -4. Questions may ask for a specific minor, a specific cofactor, or determinant evaluation using cofactor expansion. Using Mij as the cofactor without applying the sign factor.
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