91Ó°ÊÓ

First, we need to find the transpose of the given matrix. The transpose of a matrix is obtained by interchanging the rows and columns. For a given matrix A with elements \(a_{ij}\), the transpose \(A^T\) will have elements \( a_{ji}\). So, the transpose of matrix P is: $$ P^T=\left[\begin{array}{rr} -1 & 2 \\ 1 & -1 \\ 2 & -2 \end{array}\right] $$

Now, we will find the matrix multiplication of the given matrix P with its transpose P^T. To multiply two matrices, the number of columns in the first matrix should be equal to the number of rows in the second matrix. In this case, P has 3 columns and P^T has 3 rows, so it's possible to multiply them. Matrix multiplication is done by multiplying each element of a row in the first matrix with each element of a column in the second matrix and summing the result. If the first matrix is of order m×n and the second matrix is of order n×p, the resulting matrix will be of order m×p. Here, P is 2x3 and P^T is 3x2, so the resulting matrix will be a 2x2 matrix. Let's denote the resulting matrix Q: $$ Q=P×P^T=\left[\begin{array}{rr} (-1)(-1) + (1)(1) + (2)(2) & (-1)(2) + (1)(-1) + (2)(-2) \\ (2)(-1) + (-1)(1) + (-2)(2) & (2)(2) + (-1)(-1) + (-2)(-2) \end{array}\right]=\left[\begin{array}{rr} 6 & -6\\ -6 & 9 \end{array}\right] $$

Since the resulting matrix Q is a 2x2 square matrix, we can determine its determinant. The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) can be calculated using the formula \(ad - bc\). Let's compute the determinant of matrix Q: Determinant of Q = \((6)(9) - (-6)(-6) = 54 - 36 = 18\) Hence, the determinant of the resulting matrix Q is 18.