Cofactors (Practice Questions)

A cofactor of an element in a matrix is obtained by multiplying its minor by a sign determined by the element's position in the matrix.

Solved Examples

Example 1. Find the cofactor of a₁₁ in the matrix \begin{bmatrix} 1 & 3\\ 5 & 7 \end{bmatrix}

Given matrix is \begin{bmatrix} 1 & 3\\ 5 & 7 \end{bmatrix}
Minor M₁₁ = 7
Cofactor of a₁₁ = 7 × (-1)¹⁺¹ = 7

Example 2. Find the cofactor of a₁₂ in the matrix \begin{bmatrix} 1&4 & 6\\ 8 & 7 & 9 \\ 4 & 0 & 5 \end{bmatrix}

Given matrix is \begin{bmatrix} 1&4 & 6\\ 8 & 7 & 9 \\ 4 & 0 & 5 \end{bmatrix}
Minor M₁₂ = determinant of \begin{bmatrix} 8 & 9 \\ 4 & 5 \end{bmatrix}
M₁ = 40 - 36 = 4
Cofactor C₁ of a₁₂ = M₁₂ × (-1)¹⁺²
C₁₂ = 4 × (-1) = -4

Example 3. What is the cofactor matrix of \begin{bmatrix} 3 & 6\\ 7 & 8 \end{bmatrix}

Step 1: Find the minor of each element and make a minor matrix.
Minor of a₁₁ is calculated by eliminating the row 1 and column 1 as follows
M₁₁ = 8
Similarly minor of element a₁₂ is calculated by eliminating the row 1 and column 2 as follows:
M₁₂ = 7
Similarly calculate minors of all elements to obtain the following minor matrix:
\begin{bmatrix} 8 & 7\\ 6 & 3 \end{bmatrix}

Step 2: Multiply each element of the minor matrix by (-1)^i+j to get the cofactor of that element i.e. C_ij
Cofactor of M₁₁ is calculated as follows:
C₁₁ = M₁₁ × (-1)¹⁺¹ = M₁₁ = 8
Cofactor of M₁₂ is calculated as follows:
C₁₂ = M₁₂ × (-1)¹⁺² = -M₁₂ = -7
Similarly calculate the other cofactors to obtain the following cofactor matrix:
\begin{bmatrix} 8 & -7\\ -6 & 3 \end{bmatrix}

Example 4. What is the cofactor matrix of \begin{bmatrix} 0 & -6\\ -3 & -4 \end{bmatrix}

Step 1: Find the minor of each element and make a minor matrix.
M₁₁ = -4
M₁₂ = -3
Similarly calculate minors of all elements to obtain the following minor matrix:
\begin{bmatrix} -4 & -3\\ -6 & 0 \end{bmatrix}

Step 2: Multiply each element of the minor matrix by (-1)^i+j to get the cofactor of that element i.e. C_ij
C₁₁ = M₁₁ × (-1)¹⁺¹ = M₁₁ = -4
C₁₂ = M₁₂ × (-1)¹⁺² = -M₁₂ = -3
Similarly calculate the other cofactors to obtain the following cofactor matrix:
\begin{bmatrix} -4 & 3\\ 6 & 0 \end{bmatrix}

Example 5. What is the cofactor matrix of \begin{bmatrix} 2&4 & 6\\ 7 & 9&2\\ 1& 6 & 0 \end{bmatrix}

Step 1: Find the minor of each element and make a minor matrix.
M₁₁ = determinant of \begin{bmatrix} 9 & 2\\ 6 & 0 \end{bmatrix}
M₁₁ = 0 -12 = -12
M₁₂ = determinant of \begin{bmatrix} 7 & 2\\ 1 & 0 \end{bmatrix}
M₁₂ = 0 - 2 = -2
Similarly calculate minors of all elements to obtain the following minor matrix:
\begin{bmatrix} -12 & -2 & -33\\ -36 & -6 & 8\\ -46 & -38 & -10 \end{bmatrix}

Step2 : Multiply each element of the minor matrix by (-1)^i+j to get the cofactor of that element i.e. C_ij
C₁₁ = M₁₁ × (-1)¹⁺¹ = M₁₁ = -12
C₁₂ = M₁₂ × (-1)¹⁺² = -M₁₂ = 2
Similarly calculate the other cofactors to obtain the following cofactor matrix:
\begin{bmatrix} -12 & 2 & -33\\ 36 & -6 & -8\\ -46 & 38 & -10 \end{bmatrix}

Practice Problems

Problem 1: Find the cofactor of the element in the second row and third column: A = \begin{bmatrix} 2 & 3 & 4\\ 6 &6 & 5\\ 7 & 8 & -10 \end{bmatrix}.

Problem 2: Find the cofactor matrix of A = \begin{bmatrix} 12 & 7 & 4\\ 4 &1 & 5\\ 7 & 0 & 5 \end{bmatrix}.

Problem 3: Find the cofactor matrix of A = \begin{bmatrix} 4 & 0 & 1\\ 16 &2 & 9\\ 8 & 3 & 7 \end{bmatrix}.

Problem 4: Find the cofactor matrix of A = \begin{bmatrix} 1 & 0 & 0\\ 1 &0 &1\\ 4& 1 & 6 \end{bmatrix}.

Problem 5: Find the cofactor matrix of A = \begin{bmatrix} 3 & 5 & 7\\ 9 &11 & -13\\ -17 & 19 & 23 \end{bmatrix}.

Solutions of Practice Problems:

1. \begin{bmatrix}-100 & 95 & 6 \\62 & -48 & 5 \\-9 & 14 & -6\end{bmatrix}
2. \begin{bmatrix} 5 & 15 & -7\\ -35 & 32 & 49\\ 31 & -44 & -16 \end{bmatrix}
3. \begin{bmatrix} -13 & -40 & 32\\ 3 &20 & -12\\ -2 & -20 & 8 \end{bmatrix}
4. \begin{bmatrix} -1 & -2 & 1\\ 0 & 6 & -1\\ 0 & -1 & 0 \end{bmatrix}
5. \begin{bmatrix} 500 & 14 & 358\\ 18 &188 & -142\\ -142 & 102 & -12 \end{bmatrix}