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Question

Mathematics Question on Matrices

Find the inverse of each of the matrices, if it exists[21 42]\begin{bmatrix} 2 & 1 \\\ 4 & 2 \end{bmatrix}

Answer

Let A=[21 42]\begin{bmatrix} 2 & 1 \\\ 4 & 2 \end{bmatrix}

We know that A = IA

[21 42]\begin{bmatrix} 2 & 1 \\\ 4 & 2 \end{bmatrix}=[10 01]\begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix}A

Applying R1→R1−12R2R_1\rightarrow R_1-\frac{1}{2}R_2, we have:

[00 42]\begin{bmatrix} 0 & 0 \\\ 4 & 2 \end{bmatrix}=[1−12 01]\begin{bmatrix} 1 & -\frac12 \\\ 0 & 1 \end{bmatrix}A

Now, in the above equation, we can see all the zeros in the first row of the matrix on the L.H.S. Therefore, A−1 does not exist.