Which of the following matrices does not have an inverse?
[tex]\left[\begin{array}{ll}8 & 4 \\ 4 & 2\end{array}\right][/tex]
[tex]\left[\begin{array}{cc}8 & -4 \\ 2 & 1\end{array}\right][/tex]
[tex]\left[\begin{array}{ll}4 & 0 \\ 0 & 1\end{array}\right][/tex]
[tex]\left[\begin{array}{cc}2 & 2 \\ -2 & 2\end{array}\right][/tex]
Answer (2)
Calculate the determinant of each matrix.
Matrix A has a determinant of ( 8 × 2 ) − ( 4 × 4 ) = 0 .
Since the determinant of matrix A is 0, it does not have an inverse.
The matrix that does not have an inverse is [ 8 4 4 2 ] .
Explanation
Problem Analysis We are given four 2x2 matrices and asked to identify the one that does not have an inverse. A matrix has an inverse if and only if its determinant is non-zero. Therefore, we need to calculate the determinant of each matrix and find the one with a determinant of 0.
Calculating Determinants Let's calculate the determinant of each matrix.
Matrix A: [ 8 4 4 2 ] . The determinant is ( 8 × 2 ) − ( 4 × 4 ) = 16 − 16 = 0 .
Matrix B: [ 8 − 4 2 1 ] . The determinant is ( 8 × 1 ) − ( − 4 × 2 ) = 8 − ( − 8 ) = 16 .
Matrix C: [ 4 0 0 1 ] . The determinant is ( 4 × 1 ) − ( 0 × 0 ) = 4 − 0 = 4 .
Matrix D: [ 2 2 − 2 2 ] . The determinant is ( 2 × 2 ) − ( 2 × − 2 ) = 4 − ( − 4 ) = 8 .
Identifying the Matrix without an Inverse Since matrix A has a determinant of 0, it does not have an inverse. The other matrices have non-zero determinants, so they do have inverses.
Final Answer Therefore, the matrix that does not have an inverse is [ 8 4 4 2 ] .
Examples
Understanding matrix inverses is crucial in various fields like computer graphics, cryptography, and solving systems of linear equations. For instance, in computer graphics, transformations like rotations and scaling are represented by matrices. To undo a transformation, you need the inverse of the transformation matrix. Similarly, in cryptography, matrix inverses can be used to decode encrypted messages. Knowing which matrices have inverses and how to find them is a fundamental skill in these applications.
Matrix A [ 8 4 4 2 ] does not have an inverse because its determinant is 0. The other matrices have non-zero determinants, indicating that they all have inverses. Therefore, Matrix A is the correct choice.
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