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Minor

Find the mino...

Question

Find the minor of the element of second row and third column ( $a 23$ ) in the following determinant:
$∣ ∣ ∣ ∣ \begin{matrix} 2 & - 3 & 5 6 & 0 & 4 1 & 5 & - 7 \end{matrix} ∣ ∣ ∣ ∣$

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Solution

We have,
$∣ ∣ ∣ ∣ \begin{matrix} 2 & - 3 & 5 6 & 0 & 4 1 & 5 & - 7 \end{matrix} ∣ ∣ ∣ ∣$
Minor of an element $a_{23} = M_{23} = ∣ ∣ ∣ \begin{matrix} 2 & - 3 1 & 5 \end{matrix} ∣ ∣ ∣ = 10 + 3 = 13$

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Q. Which of the following is not correct in a given determinant of A, where A = [a_ij]_3×3.
(a) Order of minor is less than order of the det (A)
(b) Minor of an element can never be equal to cofactor of the same element
(c) Value of determinant is obtained by multiplying elements of a row or column by corresponding cofactors
(d) Order of minors and cofactors of elements of A is same

Q. For the determinant 2 -3 5 .
6 0 4
1 5 -7
Find M₁₂ and C₂₃ where M₁₂is a minor of the element in first row and second column & C₂₃ is a cofactor of the element in the second row and third column.

Consider the determinant

Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣

M_{i j} =

Minor of the element of

i^{t h}

j^{t h}

C_{i j} =

Cofactor of element of

i^{t h}

j^{t h}

a_{3} M_{13} - b_{3} M_{23} + c_{3} M_{33}

Consider the determinant

Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣

M_{i j} =

Minor of the element of

i^{t h}

j^{t h}

C_{i j} =

Cofactor of element of

i^{t h}

j^{t h}

a_{2} . C_{12} + b_{2} . C_{22} + c_{2} . C_{32}

Consider the determinant

Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣

M_{i j} =

Minor of the element of

i^{t h}

j^{t h}

C_{i j} =

Cofactor of element of

i^{t h}

j^{t h}

If all the elements of the determinant are multiplied by 2, then the value of new determinant is

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