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Matrix multiplication is a crucial concept in linear algebra. It involves creating a new matrix by multiplying two matrices together. For matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Here's a step-by-step guide on performing matrix multiplication:

• Take the elements in the row of the first matrix.
• Multiply them by the corresponding elements in the column of the second matrix.
• Add the products to get a single number, which is the entry in the resultant matrix.

In the problem we've worked on, you're asked to multiply matrix A by its transpose, A^T. This multiplication is crucial in finding whether a matrix is orthogonal when its product with its transpose equals the Identity Matrix.

The identity matrix is a special type of square matrix, analogous to the number 1 in multiplication. It operates as a multiplicative identity for matrices, meaning if you multiply any matrix by the identity matrix, the original matrix is unchanged.

An identity matrix has ones on its main diagonal (from the top-left to the bottom-right) and zeros elsewhere. For a 3x3 identity matrix, denoted as I₃, it looks like this:

• \[ I_3 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \]

In this exercise, the goal was to show that multiplying matrix A with its transpose results in the identity matrix, affirming that A is orthogonal. This property helps verify calculations and equipped us to solve for the specific terms that give A its orthogonality.

A matrix transpose, denoted as A^T, involves rearranging the matrix such that rows are turned into columns and vice versa. This operation is crucial in linear algebra, especially in determining orthogonal matrices, among other things.

For a given matrix A:

• The first row becomes the first column of A^T.
• The second row becomes the second column, and so forth.

Mathematically, if A is a matrix represented by \[ A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \] its transpose, A^T, will be \[ A^T = \begin{pmatrix} a & c \ b & d \end{pmatrix}. \] Transposing was an essential step in our problem when verifying that A multiplied by A^T equals the identity matrix.

The magnitude of a number often refers to its absolute value, which is essentially the number's distance from zero on a number line, disregarding any sign. This concept is particularly important when dealing with quantities that represent size or length, as magnitudes are always non-negative.

To determine the magnitude, represented as \(|a|\), simply consider a:

• If a is positive or zero, |a| is a itself.
• If a is negative, |a| is -a (the opposite number, which is positive).

In the problem at hand, we found the magnitude of the component \(|p|\) by solving the equation derived from making the matrix product equal to the identity matrix. The importance of magnitude here is that it provides a positive length of the vector, which was \[ |p| = \frac{1}{\sqrt{2}}. \] This indicates that no matter whether p is positive or negative, its absolute size is critical in maintaining orthogonality in the matrix.