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This is the second of our many quizzes on Inverse matrix. Click on this to access more such competitive quizzes on Algebra.

Inverse matrix Quiz 2

mathematics

 Vera had the matrix, its adjoint and determinant written on a piece of paper. She lost the original matrix and still has the adjoint and determinant with her. Can she find the inverse of the original matrix using this information?

 Try thinking of the formula of inverse for matrix. If supplement information like determinant and adjoint of the matrix are given, the original matrix is not needed. 

  The formula for inverse of a matrix is $ \\ {{A}^{-1}}=\dfrac{1}{\det (A)}adj(A) \\ $ Where det$(A)$ is the determinant and adj$(A)$ is the adjoint of the matrix. $ \\ $ So the inverse of the matrix can be found with determinant and adjoint alone.

 A video game stores the action of the player as a matrix, the next move by the computer is set as the inverse of the previous move. If at an instant the matrix is as $\left( \begin{matrix}   3 & 4  \\   6 & 8  \\ \end{matrix} \right)$. What would be the next move?

 Not every matrix has an inverse. The inverse of a matrix doesn’t exist if the value of the determinant is $0$.

  The original matrix is given as $ \\  \left( \begin{matrix}  3 & 4  \\  6 & 8  \\\end{matrix} \right) \\ $ Finding the inverse by swapping the positions of $a$ and $d$, putting negatives in front of $b$ and $c$, and dividing the matrix by  the determinant $\left( \text{ad-bc} \right)$ $ \\ \left( \begin{matrix}   8 & -4  \\   -6 & 3  \\\end{matrix} \right) \\ $ The determinant of the matrix is $ \\ \Rightarrow \text{(3}\times 8\text{)-(4}\times 6\text{)=24-24=0} \\ $ Dividing the matrix by determinant will not produce any outcome as determinant is $0$ and number cannot be divided by $0$.

 Brendon wants to divide a matrix $[118.4\text{  }135.2]$  with the matrix $\left( \begin{matrix}   3 & 3.5  \\   3.2 & 3.6  \\\end{matrix} \right)$. Help him in doing so.

 Matrix cannot be divided directly. To divide them multiply the inverse of dividend with the divisor.

Finding the inverse of the matrix to divide it $ \\ {{\left( \begin{matrix}   3 & 3.5  \\   3.2 & 3.6  \\ \end{matrix} \right)}^{-1}}=\dfrac{1}{Det(A)}Adj(A) \\ $ determinant of $A =(3\times 3.6\text{)-(3}\text{.5}\times 3.2\text{)=10}\text{.8-11}\text{.2=0}\text{.4} \\  Adj(A)=\left( \begin{matrix}   3.6 & -3.5  \\   -3.2 & 3  \\\end{matrix} \right) \\ $  Inverse $(A)=\left( \begin{matrix}   -9 & 8.75  \\   8 & -7.5  \\\end{matrix} \right) \\ $ Multiplying the matrix with the inverse to divide it $ \\ \Rightarrow [118.4\text{  135}\text{.2}]\left( \begin{matrix}   -9 & 8.75  \\   8 & -7.5  \\\end{matrix} \right) \\  \Rightarrow [118.4\times -9+135.2\times 8+118.4\times 8.75+135.2\times -7.5] \\  \Rightarrow [16\text{  22}]$

 A child wants to practice the concepts of matrices. For practice he decides to find the inverse of the previous obtained matrix. He reaches a stage where the inverse cannot be found anymore. What family does the matrix come under?

 Take a look at the examples of each form of matrix provided in the options and try to figure out what they mean.

 Singular matrix- A matrix is singular if the determinant of the matrix is zero. Since the determinant is $0$ it cannot be divided by the adjoint to find inverse and hece, inverse does not exist.

 Tom and Jerry like to shop from the same place. Tom bought 5 eggs and 3 chocolates for Rs. 22. Jerry bought 4 eggs and 6 chocolates for Rs 33.Find the cost of 1 egg.

 $\text{Let the cost of 1 egg be e and 1 chocolate be c} \\  \text{ }\!\![\!\!\text{ e  c }\!\!]\!\!\text{ }\left( \begin{matrix}   5 & 4  \\   2 & 6  \\ \end{matrix} \right)\text{= }\!\![\!\!\text{ 22  33 }\!\!]\!\!\text{  } \\  \text{Finding the inverse of }\left( \begin{matrix}   5 & 4  \\   2 & 6  \\\end{matrix} \right) \\  \Rightarrow \dfrac{1}{(5)(6)-(4)(2)}\left( \begin{matrix}   6 & -4  \\   -2 & 5  \\\end{matrix} \right) \\  \Rightarrow \dfrac{1}{22}\left( \begin{matrix}   6 & -4  \\   -2 & 5  \\\end{matrix} \right) \\  \text{Therefore} \\  [e\text{  }c]=\dfrac{1}{22}[22\text{  33}]\left( \begin{matrix}   6 & -4  \\   -2 & 5  \\\end{matrix} \right) \\  \Rightarrow [3\text{   }3.5] \\  \text{The corresponding value of egg is 3}$