Understanding the Product of a Square Matrix and its Transpose

The question asks about the nature of the product when a square matrix is multiplied by its transpose. Let's break down the concepts involved.

What is a Square Matrix?

A square matrix is a matrix that has the same number of rows and columns. If a matrix A has an order of $n \times n$, it is a square matrix of order n. The question specifies a square matrix of order 3, meaning it is a $3 \times 3$ matrix.

What is the Transpose of a Matrix?

The transpose of a matrix A, denoted by $A^T$ or $A'$, is obtained by interchanging the rows and columns of A. If A is an $m \times n$ matrix, then $A^T$ is an $n \times m$ matrix. If A is a square matrix of order n ($n \times n$), its transpose $A^T$ will also be a square matrix of order n ($n \times n$).

For example, if

$\text{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

Then its transpose is

$\text{A}^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$

Product of a Matrix and its Transpose ($A A^T$)

We are interested in the product of a square matrix A and its transpose $A^T$. Since A is an $n \times n$ matrix and $A^T$ is also an $n \times n$ matrix, their product $A A^T$ will be an $n \times n$ matrix.

Let $B = A A^T$. We need to determine the nature of this matrix B.

To do this, let's consider the transpose of B, which is $B^T$. We can use the property of matrix transposition that for any two matrices X and Y for which the product XY is defined, $(XY)^T = Y^T X^T$.

Applying this property to $B = A A^T$, we get:

$\text{B}^T = (A A^T)^T$

Using the property $(XY)^T = Y^T X^T$, where $X=A$ and $Y=A^T$:

$\text{B}^T = (A^T)^T A^T$

Another important property of matrix transposition is that the transpose of the transpose of a matrix is the original matrix, i.e., $(A^T)^T = A$.

Substituting $(A^T)^T = A$ into the expression for $B^T$:

$\text{B}^T = A A^T$

We defined $B = A A^T$, so we have found that:

$\text{B}^T = B$

What is a Symmetric Matrix?

A square matrix M is called a symmetric matrix if its transpose is equal to the matrix itself, i.e., $M^T = M$.

Since we found that $(A A^T)^T = A A^T$, the matrix $A A^T$ satisfies the condition for being a symmetric matrix.

Similarly, let's consider the product $C = A^T A$. Since $A^T$ is $n \times n$ and A is $n \times n$, the product $A^T A$ will also be an $n \times n$ matrix.

Let's find the transpose of C:

$\text{C}^T = (A^T A)^T$

Using the property $(XY)^T = Y^T X^T$, where $X=A^T$ and $Y=A$:

$\text{C}^T = A^T (A^T)^T$

Using $(A^T)^T = A$:

$\text{C}^T = A^T A$

We defined $C = A^T A$, so we have found that:

$\text{C}^T = C$

Thus, the product $A^T A$ is also a symmetric matrix.

Evaluating the Options

Based on our analysis, the product of a square matrix and its transpose (either $A A^T$ or $A^T A$) is always a symmetric matrix.

Let's look at the given options:

1. Unit Matrix: A unit matrix is a specific type of diagonal matrix with 1s on the main diagonal and 0s elsewhere. $A A^T$ or $A^T A$ is not always a unit matrix. This is only true under specific conditions, like when A is an orthogonal matrix.
2. Zero Matrix: A zero matrix is a matrix where all entries are zero. $A A^T$ or $A^T A$ is not always a zero matrix unless A is the zero matrix itself.
3. Identity Matrix: The identity matrix is another name for the Unit matrix. The same reasoning as for the Unit Matrix applies. $A A^T$ or $A^T A$ is not always an identity matrix.
4. Symmetric Matrix: As shown above, the product of a square matrix A and its transpose $A^T$ (i.e., $A A^T$) is always a symmetric matrix.

Therefore, the product of a square matrix of order 3 (or any order n) and its transpose is a symmetric matrix.

Conclusion

The product of a square matrix of order 3 and its transpose is a symmetric matrix because $(A A^T)^T = A A^T$.

Revision Table: Key Matrix Types

Additional Information: Properties of Symmetric Matrices

• If A and B are symmetric matrices of the same order, then $A+B$ and $A-B$ are also symmetric matrices.
• If A is a symmetric matrix and k is a scalar, then kA is also a symmetric matrix.
• If A and B are symmetric matrices of the same order, then AB is symmetric if and only if AB = BA (i.e., A and B commute).
• For any matrix A (not necessarily square), the matrices $A A^T$ and $A^T A$ are always symmetric.
• Every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix. A matrix M is skew-symmetric if $M^T = -M$.