https://file.aiquickdraw.com/mathbot/file/1726910570909buvjvngr.jpg

The problem asks to prove that:

\[
\begin{vmatrix}
a^2 + b^2 & ac + ad \\
ac + ad & c^2 + d^2
\end{vmatrix}
=
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}^2
\]

Let's go step by step to prove this.

### Step 1: Calculate the determinant of the left-hand side matrix

The determinant of a 2x2 matrix:
\[
\begin{vmatrix}
p & q \\
r & s
\end{vmatrix}
= ps - qr
\]

For the matrix:
\[
\begin{vmatrix}
a^2 + b^2 & ac + ad \\
ac + ad & c^2 + d^2
\end{vmatrix}
\]

The determinant is:
\[
\text{Determinant} = (a^2 + b^2)(c^2 + d^2) - (ac + ad)^2

Prove that the determinant of the matrix [[a^2 + b^2, ac + ad], [ac + ad, c^2 + d^2]] is equal to the square of the determinant of the matrix [[a, b], [c, d]].

Explore the detailed step-by-step proof showing that the determinant of the matrix [[a^2 + b^2, ac + ad], [ac + ad, c^2 + d^2]] is equal to the square of the determinant of the matrix [[a, b], [c, d]]. This proof covers key concepts in linear algebra, such as determinants and matrix operations.

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