If A = [[1, 2], [3, 4]], find det(A) and adj(A).
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Step-by-Step Solution
Step 1: Calculate the Determinant of A
For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is calculated as the product of the elements on the main diagonal minus the product of the elements on the anti-diagonal. Here, $a=1$, $b=2$, $c=3$, and $d=4.
Step 2: Substitute values and compute determinant
Substitute the values $a=1$, $b=2$, $c=3$, and $d=4$ into the determinant formula. This gives us (1)(4) - (2)(3)$, which simplifies to $4 - 6 = -2.
Step 3: Calculate the Adjoint of A
For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the adjoint matrix is found by swapping the elements on the main diagonal ($a$ and $d$) and changing the signs of the elements on the anti-diagonal ($b$ and $c$).
Step 4: Substitute values into the adjoint formula
Substitute the values $a=1$, $b=2$, $c=3$, and $d=4$ into the adjoint formula. The element $d$ becomes the top-left, $a$ becomes the bottom-right, $-b$ becomes the top-right, and $-c$ becomes the bottom-left.