Answer

Verified

418.2k+ views

**Hint:**We have given a $3 \times 3$ matrix we have to calculate its inverse. So firstly we have to calculate the determinant of the matrix if the determinant is equal to zero. Then the inverse of the matrix does not exist. If the determinant is not equal to zero then the inverse of the matrix exists. After we calculate the cofactor of each row and then we calculate the adjoint of \[A\]. Adjoint of \[A\] can be calculated by putting the co-factors in a matrix and taking its transpose then we apply formula of inverse of the matrix

Inverse of $A = {A^{ - 1}} = \dfrac{{adjA}}{{\left| A \right|}}$

Here \[adjA\] is adjoint of \[A\] and $\left| A \right|$ is determinant of \[A\].

**Complete step-by-step answer:**

We have been given a matrix

$A = \left[ {\begin{array}{*{20}{c}}

1&{ - 1}&2 \\

{ - 2}&1&3 \\

4&{ - 2}&2

\end{array}} \right]$

we calculate its determinant,

$\left| A \right| = \left[ {\begin{array}{*{20}{c}}

1&{ - 1}&2 \\

{ - 2}&1&3 \\

4&{ - 2}&2

\end{array}} \right]$

$ = 1(2 + 6) + 1( - 4 - 12) + 2(4 - 4)$

$ = 1(8) + 1( - 16) + 2(0)$

$ \Rightarrow 8 - 16 + 0 = - 8$

$\left| A \right| = - 8 \ne 0$

So, ${A^{ - 1}}$ exist

Now we calculate cofactors of Matrix $A$

Cofactor of first row is given as :

$\left| {\begin{array}{*{20}{c}}

1&3 \\

{ - 2}&2

\end{array}} \right|, - \left| {\begin{array}{*{20}{c}}

{ - 2}&3 \\

4&2

\end{array}} \right|,\left| {\begin{array}{*{20}{c}}

{ - 2}&1 \\

4&{ - 2}

\end{array}} \right| = - 8,16,0$

Cofactor of second row is given as:

$ - \left| {\begin{array}{*{20}{c}}

{ - 1}&2 \\

{ - 2}&2

\end{array}} \right|,\left| {\begin{array}{*{20}{c}}

1&2 \\

4&2

\end{array}} \right|, - \left| {\begin{array}{*{20}{c}}

1&{ - 1} \\

4&{ - 2}

\end{array}} \right| = - 2, - 6, - 2$

Cofactor of Third row is given as:

\[\left| {\begin{array}{*{20}{c}}

{ - 1}&2 \\

1&3

\end{array}} \right|, - \left| {\begin{array}{*{20}{c}}

1&2 \\

{ - 2}&3

\end{array}} \right|,\left| {\begin{array}{*{20}{c}}

1&{ - 1} \\

{ - 2}&1

\end{array}} \right| = - 5, - 7, - 1\]

Now, \[adjA = {\left[ {\begin{array}{*{20}{c}}

8&{16}&0 \\

{ - 2}&{ - 6}&{ - 2} \\

{ - 5}&{ - 7}&{ - 1}

\end{array}} \right]^1} = \left[ {\begin{array}{*{20}{c}}

8&{ - 2}&{ - 5} \\

{16}&{ - 6}&{ - 7} \\

0&{ - 2}&{ - 1}

\end{array}} \right]\]

Adjoint of \[A = adjA = \left[ {\begin{array}{*{20}{c}}

8&{ - 2}&{ - 5} \\

{16}&{ - 6}&{ - 7} \\

0&{ - 2}&{ - 1}

\end{array}} \right]\]

${A^{ - 1}} = \dfrac{{adjA}}{{\left| A \right|}} = \dfrac{1}{{ - 8}}\left[ {\begin{array}{*{20}{c}}

8&{ - 2}&{ - 5} \\

{16}&{ - 6}&{ - 7} \\

0&{ - 2}&{ - 1}

\end{array}} \right]$

\[{A^{ - 1}} = \left[ {\begin{array}{*{20}{c}}

{ - 1}&\dfrac{1}{{4}}&\dfrac{5}{{ 8}} \\

{ - 2}&\dfrac{3}{{4}}&\dfrac{7}{{8}} \\

0&\dfrac{1}{{4}}&\dfrac{1}{{ 8}}

\end{array}} \right]\]

**Note:**In mathematics a matrix is a rectangular array or table of numbers symbols, or expressions, arranged in row and column. Matrices with $m$ rows and $n$ columns are called $m \times n$ matrices. Matrices can be used to compactly write and work with multiple linear questions, that is a system of linear equations. A square matrix has an inverse if its determinant is not equal to zero. Such matrices are also called invertible matrices.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE