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find the non singular matrices P and Q such rhat PAQ is of the normal form and find its rank

The given image represents a matrix. Let's start by rewriting the matrix and identifying its form.

The matrix provided is:
\[
A = \begin{bmatrix} 
12 & 3 & -2 \\ 
2 & -2 & 13 \\ 
3 & 0 & 41 
\end{bmatrix}
\]

To find non-singular matrices \(P\) and \(Q\) such that \(PAQ\) is in normal form (often reduced to its row echelon or reduced row echelon form), we perform row and column operations. The rank of the matrix is the number of non-zero rows in its reduced row echelon form.

### Step 1: Row Reduction to Row Echelon Form

Let's perform Gaussian elimination to reduce matrix \(A\) to its row echelon form. This will help us identify its rank.

I'll compute the row echelon form of the matrix.

Find the non-singular matrices P and Q such that PAQ is of the normal form and find its rank.

Solve the problem of finding non-singular matrices P and Q such that PAQ is in normal form and determine the matrix's rank. This solution involves concepts of matrix algebra, including Gaussian elimination and row echelon form. Suitable for undergraduate math students.

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