### Determinants Exercise 04

## Question:

Write minors and cofactors of the elements of following determinants (i)$\left|\begin{array}{cc}2& -4\\ 0& 3\end{array}\right|$ (ii)$\left|\begin{array}{cc}\mathrm{a}& \mathrm{c}\\ \mathrm{b}& \mathrm{d}\end{array}\right|$

## Answer:

Answer: (i) Minors: M11 = 3, M12 = -4, M21 = 0, M22 = 2 Cofactors: C11 = -3, C12 = 4, C21 = 0, C22 = -2

(ii) Minors: M11 = d, M12 = -c, M21 = -b, M22 = a Cofactors: C11 = -d, C12 = c, C21 = b, C22 = -a

## Question:

Find the Minors and Cofactors of the elements of the following determinants: (i)$\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$ (ii)$\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -2\\ 0& 1& 2\end{array}\right|$

## Answer:

(i) Minor of 1 = 1; Cofactor of 1 = 1 Minor of 0 = 0; Cofactor of 0 = 0 Minor of 0 = 0; Cofactor of 0 = 0

(ii) Minor of 1 = -20; Cofactor of 1 = -20 Minor of 0 = 0; Cofactor of 0 = 0 Minor of 4 = -8; Cofactor of 4 = 8 Minor of 3 = 10; Cofactor of 3 = -10 Minor of 5 = -2; Cofactor of 5 = 2 Minor of -2 = 10; Cofactor of -2 = -10 Minor of 0 = 0; Cofactor of 0 = 0 Minor of 1 = -20; Cofactor of 1 = 20 Minor of 2 = 8; Cofactor of 2 = -8

## Question:

Using cofactors of elements of second row, evaluate Δ= $\left|\begin{array}{ccc}5& 3& 8\\ 2& 0& 1\\ 1& 2& 3\end{array}\right|$

## Answer:

Step 1: Calculate the cofactors of the elements of the second row.

Cofactor of 2 = (−3)

Cofactor of 0 = 2

Cofactor of 1 = (−1)

Step 2: Multiply the cofactors with the elements of the second row and add them up.

(−3) × 2 + 2 × 0 + (−1) × 1 = (−3) + 0 + (−1) = −4

Step 3: Calculate the determinant by multiplying the sum from Step 2 with the determinant of the original matrix.

Δ = (−4) × $\left|\begin{array}{ccc}5& 3& 8\\ 2& 0& 1\\ 1& 2& 3\end{array}\right|$

Δ = (−4) × (−14) = 56

## Question:

If Δ=$\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|$ and Aij is cofactors of aij, then the value of Δ is given by A a11A31+a12A32+a13A33 B a11A11+a12A21+a13A31 C a21A11+a22A12+a23A13 D a11A11+a21A21+a31A31

## Answer:

A. A11A31 + a12A32 + a13A33 B. a11A11 + a12A21 + a13A31 C. a21A11 + a22A12 + a23A13 D. a11A11 + a21A21 + a31A31

## Question:

Using cofactors of elements of third column evaluate Δ= $\left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{yz}\\ 1& \mathrm{y}& \mathrm{zx}\\ 1& \mathrm{z}& \mathrm{xy}\end{array}\right|$

## Answer:

Step 1: Find the cofactors of elements of third column.

C11 = yz, C12 = -zx, C13 = xy

Step 2: Evaluate Δ using the cofactors.

Δ = |1 x yz| |1 y -zx| |1 z xy|

Δ = yz(-zx) + (-zx)xy + xy(yz)

Δ = -yzxz + xyz2