Write the minors and cofactors of each element of the first column of the matrices and hence evaluate the determinant in each case:

$$A=\begin{bmatrix}1 &a&bc \\[0.3em] 1 & b&ca \\[0.3em] 1 &c&ab \\[0.3em] \end{bmatrix}$$

Asked by Aaryan | 1 year ago |  186

##### Solution :-

Let Mij and Cij represents the minor and co–factor of an element, where i and j represent the row and column. The minor of the matrix can be obtained for a particular element by removing the row and column where the element is present. Then finding the absolute value of the matrix newly formed.

Also, Cij = (–1)i+j × Mij

$$M_{11}=ab^2-ac^2$$

$$M_{21}=a^2b-c^2b$$

M31 = a × c a – b × bc

M31 = a2c – b2c

C11 = (–1)1+1 × M11

= 1 × (ab2 – ac2)

= ab2 – ac2

C21 = (–1)2+1 × M21

= –1 × (a2b – c2b)

= c2b – a2b

C31 = (–1)3+1 × M31

= 1 × (a2c – b2c)

= a2c – b2c

Now expanding along the first column we get

|A| = a11 × C11 + a21× C21+ a31× C31

= 1× (ab2 – ac2) + 1 × (c2b – a2b) + 1× (a2c – b2c)

= ab2 – ac2 + c2b – a2b + a2c – b2c

Answered by Sakshi | 1 year ago

### Related Questions

#### \begin{vmatrix}2&3&-5 \\[0.3em] 7& 1&-2\\[0.3em] -3&4&1\end{vmatrix}

Evaluate

$$\begin{vmatrix}2&3&-5 \\[0.3em] 7& 1&-2\\[0.3em] -3&4&1\end{vmatrix}$$

#### Show that \begin {vmatrix}sin10°&-cos10° \\[0.3em] sin80°& cos80°\\[0.3em] \end{vmatrix}

Show that

$$\begin{vmatrix}sin10°&-cos10° \\[0.3em] sin80°& cos80°\\[0.3em] \end{vmatrix}$$

#### Evaluate the determinants: \begin{vmatrix}2&3&7 \\[0.3em] 13& 17&5\\[0.3em] 15&20&12\end{vmatrix}

Evaluate the determinants:

$$\begin{vmatrix}2&3&7 \\[0.3em] 13& 17&5\\[0.3em] 15&20&12\end{vmatrix}^2$$

#### Evaluate the determinants: \begin{bmatrix}a+ib &c+id \\[0.3em] -c+id & a-ib \\[0.3em]

Evaluate the determinants:

$$\begin{bmatrix}a+ib &c+id \\[0.3em] -c+id & a-ib \\[0.3em] \end{bmatrix}$$

$$\begin{bmatrix}cos 15° &-sin 15° \\[0.3em] sin75° & cos75° \\[0.3em] \end{bmatrix}$$