Which matrix is the inverse of $\left[\begin{array}{ll}3 & 7 \\ 1 & 2\end{array}\right]$ ?
A. $\left[\begin{array}{cc}-2 & -7 \\ 1 & -3\end{array}\right]$
B. $\left[\begin{array}{cc}2 & -7 \\ -1 & 3\end{array}\right]$
C. $\left[\begin{array}{cc}-2 & 7 \\ 1 & 3\end{array}\right]$
D. $\left[\begin{array}{cc}-2 & 7 \\ 1 & -3\end{array}\right]$
Answer (1)
Calculate the determinant of the matrix: d e t ( A ) = ( 3 ) ( 2 ) − ( 7 ) ( 1 ) = − 1 .
Apply the inverse formula: A − 1 = d e t ( A ) 1 [ d − b − c a ] .
Substitute the values: A − 1 = − 1 1 [ 2 − 7 − 1 3 ] .
The inverse matrix is: [ − 2 7 1 − 3 ] .
Explanation
Problem Analysis We are given a 2x2 matrix and asked to find its inverse from the given options. Let's denote the given matrix as A = [ 3 7 1 2 ] .
Inverse Formula The inverse of a 2x2 matrix [ a b c d ] is given by the formula: A − 1 = a d − b c 1 [ d − b − c a ]
Calculate Determinant First, we need to calculate the determinant of matrix A, which is d e t ( A ) = ( 3 ) ( 2 ) − ( 7 ) ( 1 ) = 6 − 7 = − 1 .
Calculate the Inverse Now, we can find the inverse of A using the formula: A − 1 = − 1 1 [ 2 − 7 − 1 3 ] = [ − 2 7 1 − 3 ]
Final Answer Comparing the calculated inverse with the given options, we find that the correct answer is [ − 2 7 1 − 3 ] .
Examples
In cryptography, matrix inverses can be used to decode messages that were encoded using a matrix. For example, if a message is encoded by multiplying it by a matrix, the recipient can decode the message by multiplying it by the inverse of the matrix. This ensures secure communication where only those with the inverse matrix can decipher the original message. This method is a basic example of how linear algebra concepts are applied in real-world security systems.
