Find co-factors of the matrix- A- - 1 0 0- 0 cos- sin- 0 sin- -cos- -

Let A= [ 1 amp; 0 amp; 0; 0 amp; cosα #160; amp; sinα #160;; 0 amp; sinα #160; amp; cosα #160; ] ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Equality of Matrices

Find co-facto...

Question

Find co-factors of the matrix,
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & cos α & sin α 0 & sin α & - cos α \end{matrix} ⎤ ⎥ ⎦$

Open in App

Solution

Let $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & cos α & sin α 0 & sin α & - cos α \end{matrix} ⎤ ⎥ ⎦$

Here, $| A | = - {cos}^{2} α - {sin}^{2} α$
$\Rightarrow | A | = - 1$ ( $∵ {sin}^{2} α + {cos}^{2} α = 1$ )
Since, $| A | \neq 0$
Hence, $A^{- 1}$ exists.

We have $A^{- 1} = \frac{a d j A}{| A |}$
and $a d j A = C^{T}$

So, we will find the co-factors of each element of $A$ .
$C_{11} = (- 1)^{1 + 1} ∣ ∣ ∣ \begin{matrix} cos α & sin α sin α & - cos α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{11} = - ({sin}^{2} α + {cos}^{2} α) = - 1$

$C_{12} = (- 1)^{1 + 2} ∣ ∣ ∣ \begin{matrix} 0 & sin α 0 & - cos α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{12} = - (0 - 0) = 0$

$C_{13} = (- 1)^{1 + 3} ∣ ∣ ∣ \begin{matrix} 0 & cos α 0 & sin α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{13} = 0 - 0 = 0$

$C_{21} = (- 1)^{2 + 1} ∣ ∣ ∣ \begin{matrix} 0 & 0 sin α & - cos α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{21} = - (0 - 0) = 0$

$C_{22} = (- 1)^{2 + 2} ∣ ∣ ∣ \begin{matrix} 1 & 0 0 & - cos α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{22} = (- cos α - 0) = - cos α$

$C_{23} = (- 1)^{2 + 3} ∣ ∣ ∣ \begin{matrix} 1 & 0 0 & sin α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{23} = - (sin α - 0) = - sin α$

$C_{31} = (- 1)^{3 + 1} ∣ ∣ ∣ \begin{matrix} 0 & 0 cos α & sin α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{31} = (0 - 0) = 0$

$C_{32} = (- 1)^{3 + 2} ∣ ∣ ∣ \begin{matrix} 1 & 0 0 & sin α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{32} = - (sin α - 0) = - sin α$

$C_{33} = (- 1)^{3 + 3} ∣ ∣ ∣ \begin{matrix} 1 & 0 0 & cos α \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{33} = (cos α - 0) = cos α$

Hence, the co-factor matrix of A is
$C = ⎡ ⎢ ⎣ \begin{matrix} - 1 & 0 & 0 0 & - cos α & - sin α 0 & - sin α & cos α \end{matrix} ⎤ ⎥ ⎦$

$\Rightarrow a d j A = C^{T} = ⎡ ⎢ ⎣ \begin{matrix} - 1 & 0 & 0 0 & - cos α & - sin α 0 & - sin α & cos α \end{matrix} ⎤ ⎥ ⎦$

Now, $A^{- 1} = \frac{a d j A}{| A |} = \frac{1}{- 1} ⎡ ⎢ ⎣ \begin{matrix} - 1 & 0 & 0 0 & - cos α & - sin α 0 & - sin α & cos α \end{matrix} ⎤ ⎥ ⎦$

$\Rightarrow A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & cos α & sin α 0 & sin α & - cos α \end{matrix} ⎤ ⎥ ⎦$

Suggest Corrections

Similar questions

Q. Evaluate :

Δ = ∣ ∣ ∣ ∣ \begin{matrix} 0 & s i n α & - c o s α - s i n α & 0 & s i n β c o s α & - s i n β & 0 \end{matrix} ∣ ∣ ∣ ∣

Q. Find co-factors of the matrix,

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 2 0 & 2 & - 3 3 & - 2 & 4 \end{matrix} ⎤ ⎥ ⎦

Q. If A= [

a_{i j}

] is a matrix of

2 \times 2

order such that

| A | = - 15

and

C_{i j}

represents the co-factors of

a_{i j}

then find the value of

a_{21} C_{21} + a_{22} C_{22}

$I f A = [\begin{matrix} c o s^{2} α & c o s α s i n α c o s α s i n α & s i n^{2} α \end{matrix}] a n d B = [\begin{matrix} c o s^{2} β & c o s β s i n β c o s β s i n β & s i n^{2} β \end{matrix}]$ are two matrices such that AB is the null matrix, then

Q. If

A = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]

is identity matrix, then write the value of α.

Join BYJU'S Learning Program

Find co-factors of the matrix,A=⎡⎢⎣1000cosαsinα0sinα−cosα⎤⎥⎦

Find co-factors of the matrix,
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & cos α & sin α 0 & sin α & - cos α \end{matrix} ⎤ ⎥ ⎦$