Find the identity matrix for the 2x2 matrix A = \[\begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}\]
Solution
Step 1 ：First, let's write down the 2x2 identity matrix I = \[\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\]
Step 2 ：Next, we need to find the matrix A^-1 such that AA^-1 = I. The formula for the inverse of a 2x2 matrix A = \[\begin{pmatrix} a & b \\ c & d \end{pmatrix}\] is A^-1 = (1/(ad - bc))\[\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\]
Step 3 ：Substitute the values from matrix A into the formula: A^-1 = (1/(4*1 - 3*2))\[\begin{pmatrix} 1 & -3 \\ -2 & 4 \end{pmatrix}\] = -2\[\begin{pmatrix} 1 & -3 \\ -2 & 4 \end{pmatrix}\] = \[\begin{pmatrix} -2 & 6 \\ 4 & -8 \end{pmatrix}\]
Step 4 ：Check that AA^-1 = I: \[\begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}\]\[\begin{pmatrix} -2 & 6 \\ 4 & -8 \end{pmatrix}\] = \[\begin{pmatrix} (-2*4 + 3*4) & (4*6 + 3*-8) \\ (-2*2 + 1*4) & (2*6 + 1*-8) \end{pmatrix}\] = \[\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\]
