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Maths > Determinants > Minors and Cofactors of Determinant
Determinants

Minors and Cofactors of Determinant

We learned how important are matrices and determinants and also studied about their wide applications. The knowledge of Minors and Cofactors is compulsory in the computation of adjoint of a matrix and hence in its inverse as well as in the computation of determinant of a square matrix. This technique of computing determinant is known as Cofactor Expansion. Let’s get started!

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Minor of a Determinant

A minor is defined as a value computed from the determinant of a square matrix which is obtained after crossing out a row and a column corresponding to the element that is under consideration. Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij.


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Cofactor of a Determinant

The cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aij is defined by A = (–1)i+j M, where M is minor of aij.

Note

  • We note that if the sum i+j is even, then Aij Mij, and that if the sum is odd, then Aij Mij.
  • Hence, the only difference between the related minor entries and cofactors may be a sign change or nothing at all.
  • Whether  or Aij Mior Aij Mij
  • has a pattern for square matrices as illustrated:
Cofactor
For example C12 M12. Of course, if you forget, you can always use the formula Ci(1)i+Mij,
Here, C12=(1)1+Mi(1)MiMij

Example

Find the minors and cofactors of all the elements of the determinant $$\begin{vmatrix} 1 & -2 \\ 4 & 3 \end{vmatrix}$$

Solution: Minor of the element aij is Mij.
Here a11 = 1. So M11 = Minor of a11 = 3
M12 = Minor of the element a12 = 4
M21 = Minor of the element a21 = –2
M22 = Minor of the element a22 = 1

Now, cofactor of aij is Aij. So,
A11 = (–1)1+1, M11 = (–1)2 (3) = 3
A12 = (–1)1+2, M12 = (–1)3 (4) = –4
A21 = (–1)2+1, M21 = (–1)3 (–2) = 2
A22 = (–1)2+2, M22 = (–1)4 (1) = 1

Solved Examples for You

Question 1: Let A=[aij]n×n be a square matirx and let cij be cofactor of aij in A. If C=[cij], then

  1. |A| = |C|
  2. |C| = |A|n-1
  3. |C| = |A|n-2
  4. none of these

Answer : We know that adjCT where C is the cofactor matrix of A.
Also |AdA|=|A|n1
Now |CT| = |Adj A|
=|A|n-1 where n it the order of the square matrix.

 Question 2: The minors and cofactors of -4 and 9 in determinant $$\begin{vmatrix} -1 & 2 & 3 \\ -4 & 5 & -6 \\ -7 & -8 & 9 \end{vmatrix}$$ are respectively
  1. 42, 42; 3, 3
  2. 42, -42; 3, 3
  3. 42, -42; 3, -3
  4. 42, 3; 42, 3

Answer : Minor of -4 is $$\begin{vmatrix} 2 & 3 \\ -8 & 9 \end{vmatrix} = 42$$
Cofactor of -4 is (1)1+2(4242

Minor of 9 is $$\begin{vmatrix} -1 & 2 \\ -4 & 5 \end{vmatrix} = 3$$
Cofactor of 9 is (1)3+.(33. Therefore, the answer is option B

Question 3: What is meant by cofactor of a matrix?

Answer: A cofactor refers to the number you attain on removing the column and row of a particular element existing in a matrix.

Question 4: What is meant by a minor matrix?

Answer: A minor refers to the square matrix’s determinant whose formation takes place by deleting one column and one row from some larger square matrix.

Question 5: Can we say that the adjoint is the same as the reverse?

Answer: The adjoint of a matrix is also known as the adjugate of a matrix. It refers to the transpose of the cofactor matrix of that particular matrix. For a matrix A, the denotation of adjoint is as adj (A). On the other hand, the inverse of a matrix A refers to a matrix which on multiplication by matrix A, results in an identity matrix.

Question 6: What is meant by rank of a matrix?

Answer: The rank of a matrix refers to the maximum number of linearly independent column vectors that exist in the matrix. Furthermore, it is also the maximum number of linearly independent row vectors that exist in the matrix.

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jannat

Is determinant available just for square matrix?

Stephen
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Stephen

Yes, it is only defined for square matrices.

J. A. Zahálka
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J. A. Zahálka

It actually is. It is equal to zero for all non-square matrices.

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