Question

Q2. If A= (1 2 6),
4 11 7
9 13 3

(a) Find the minors of 1,2 and 6.
(b) Find the cofactors of 1,2 and 6.
(c) Evaluate |A|.
(d) A^-1

Answer 1

Answer on Question #85224 – Math – Linear Algebra

Question

Q2. If $A = \begin{pmatrix} 1 & 2 & 6 \\ 4 & 11 & 7 \\ 9 & 13 & 3 \end{pmatrix}$

(a) Find the minors of 1, 2 and 6.

(b) Find the cofactors of 1, 2 and 6.

(c) Evaluate $|A|$.

(d) $A^{-1}$

Solution

$$A = \begin{pmatrix} 1 & 2 & 6 \\ 4 & 11 & 7 \\ 9 & 13 & 3 \end{pmatrix}$$

(a) Find the minors of 1, 2 and 6.

Minor of 1 is $M_{11} = \begin{vmatrix} 11 & 7 \\ 13 & 3 \end{vmatrix} = 11 \cdot 3 - 7 \cdot 13 = -58$

Minor of 2 is $M_{12} = \begin{vmatrix} 4 & 7 \\ 9 & 3 \end{vmatrix} = 4 \cdot 3 - 7 \cdot 9 = -51$

Minor of 6 is $M_{13} = \begin{vmatrix} 4 & 11 \\ 9 & 13 \end{vmatrix} = 4 \cdot 13 - 11 \cdot 9 = -47$

(b) Find the cofactors of 1, 2 and 6.

Cofactor of $a_{ij} = (-1)^{i+j} M_{ij}$

$$C_{11} = (-1)^{1+1} M_{11} = M_{11} = -58$$ $$C_{12} = (-1)^{1+2} M_{12} = -M_{12} = 51$$ $$C_{13} = (-1)^{1+3} M_{13} = M_{13} = -47$$

(c) Evaluate $|A|$.

$$|A| = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}$$ $$|A| = 1 \cdot (-58) + 2 \cdot 51 + 6 \cdot (-47) = -238$$

(d) $A^{-1}$

We can find inverse matrix by using formula

$$A^{-1} = \frac{1}{|A|} C^T$$

where $C$ is a cofactor matrix

$$C = \left( \begin{array}{ccc} C _ {1 1} & C _ {1 2} & C _ {1 3} \\ C _ {2 1} & C _ {2 2} & C _ {2 3} \\ C _ {3 1} & C _ {3 2} & C _ {3 3} \end{array} \right)$$

Find cofactors of all the elements

$$C _ {2 1} = (- 1) ^ {2 + 1} \left| \begin{array}{cc} 2 & 6 \\ 13 & 3 \end{array} \right| = -(2 \cdot 3 - 6 \cdot 13) = 72$$ $$C _ {2 2} = (- 1) ^ {2 + 2} \left| \begin{array}{cc} 1 & 6 \\ 9 & 3 \end{array} \right| = (1 \cdot 3 - 6 \cdot 9) = - 51$$ $$C _ {2 3} = (- 1) ^ {2 + 3} \left| \begin{array}{cc} 1 & 2 \\ 9 & 13 \end{array} \right| = -(13 - 2 \cdot 9) = 5$$ $$C _ {3 1} = (- 1) ^ {3 + 1} \left| \begin{array}{cc} 2 & 6 \\ 11 & 7 \end{array} \right| = (2 \cdot 7 - 6 \cdot 11) = - 52$$ $$C _ {3 2} = (- 1) ^ {3 + 2} \left| \begin{array}{cc} 1 & 6 \\ 4 & 7 \end{array} \right| = -(7 - 6 \cdot 4) = 17$$ $$C _ {3 3} = (- 1) ^ {3 + 3} \left| \begin{array}{cc} 1 & 2 \\ 4 & 11 \end{array} \right| = (11 - 2 \cdot 4) = 3$$

Construct Cofactor Matrix

$$C = \left( \begin{array}{ccc} C _ {1 1} & C _ {1 2} & C _ {1 3} \\ C _ {2 1} & C _ {2 2} & C _ {2 3} \\ C _ {3 1} & C _ {3 2} & C _ {3 3} \end{array} \right) = \left( \begin{array}{ccc} - 5 8 & 5 1 & - 4 7 \\ 7 2 & - 5 1 & 5 \\ - 5 2 & 1 7 & 3 \end{array} \right)$$

Transpose of the cofactor matrix (adjugate matrix)

$$C ^ {T} = \left( \begin{array}{ccc} - 5 8 & 7 2 & - 5 2 \\ 5 1 & - 5 1 & 1 7 \\ - 4 7 & 5 & 3 \end{array} \right)$$ $$\text{Thus } A ^ {- 1} = \frac {- 1}{238} \left( \begin{array}{ccc} - 5 8 & 7 2 & - 5 2 \\ 5 1 & - 5 1 & 1 7 \\ - 4 7 & 5 & 3 \end{array} \right) = \left( \begin{array}{ccc} 5 8 / 238 & - 7 2 / 238 & 5 2 / 238 \\ - 5 1 / 238 & 5 1 / 238 & - 1 7 / 238 \\ 4 7 / 238 & - 5 / 238 & - 3 / 238 \end{array} \right)$$

**Answer:**

(a) The minors of 1,2 and 6 are $M_{11} = -58$, $M_{12} = -51$, $M_{13} = -47$

(b) The cofactors of 1,2 and 6 are $C_{11} = -58$, $C_{12} = 51$, $C_{13} = -47$

(c) $|A| = -238$

(d) $A^{-1} = \left( \begin{array}{ccc}58 / 238 & -72 / 238 & 52 / 238 \\ -51 / 238 & 51 / 238 & -17 / 238 \\ 47 / 238 & -5 / 238 & -3 / 238 \end{array} \right).$

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