If $A = [[1, 2], [3, 4]]$ find $\det(A)$ and $\text{adj}(A)$.

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.

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