Define a null matrix and provide an example.

A null matrix (or zero matrix) is a matrix in which every element is zero.
An example of a $2 \times 3$ null matrix is:

A diagonal matrix is a square matrix in which all the elements outside the main diagonal are zero.
An example of a $3 \times 3$ diagonal matrix is:

A square matrix $A$ is classified as skew-symmetric if its transpose is equal to its negative, i.e., $A^T = -A$.

If $A^T = -A$, the matrix $A$ is skew-symmetric. For a skew-symmetric matrix, the elements on the main diagonal must be zero. This is because for any diagonal element $a_{ii}$, the condition $a_{ii} = -a_{ii}$ must hold, which implies $2a_{ii} = 0$, so $a_{ii} = 0$.
The correct option is C.
The elements on the diagonal of the matrix are equal to $\boxed{\text{C. 0}}$

We need to identify the false statement.

• A. If $|A| = 0$, then $|adj(A)| = 0$. This statement is true. If a matrix is singular ($|A|=0$), its adjoint is also singular ($|adj(A)|=0$).
• B. Adjoint of a diagonal matrix of order $3 \times 3$ is a diagonal matrix. This statement is true. The cofactors of off-diagonal elements of a diagonal matrix are zero, so its adjoint will also be diagonal.
• C. Product of two upper triangular matrices is an upper triangular matrix. This statement is true. This is a known property of triangular matrices.
• D. $adj(AB) = adj(A)adj(B)$. This statement is false. The correct property is $adj(AB) = adj(B)adj(A)$. The false statement is $\boxed{\text{D. } adj(AB) = adj(A)adj(B)}$

Given $AB = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 1 \\ 2 & 0 \end{bmatrix}$. We need to find $A^{-1}$.
Let $C = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$. So $AB = C$.
To find $A^{-1}$, we can multiply both sides by $C^{-1}$ on the right:
$A = C B^{-1}$
Then $A^{-1} = (C B^{-1})^{-1} = (B^{-1})^{-1} C^{-1} = B C^{-1}$.

First, calculate $C^{-1}$:
The determinant of $C$ is $|C| = (3)(1) - (2)(1) = 3 - 2 = 1$.
The inverse of $C$ is $C^{-1} = \frac{1}{|C|} \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix} = \frac{1}{1} \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix}$.

Now, calculate $A^{-1} = B C^{-1}$:

The correct option is C.
$A^{-1}$ is $\boxed{\text{C. } \begin{bmatrix} 0 & 1 \\ 2 & -4 \end{bmatrix}}$

Given $AB' + BA' = 0$ and $A$ is skew-symmetric.
If $A$ is skew-symmetric, then $A^T = -A$. (Note: $A'$ denotes the transpose, so $A' = A^T$).
Substitute $A^T = -A$ into the given equation:
$AB^T + B(-A) = 0$
$AB^T - BA = 0$
$AB^T = BA$

Now, we need to classify $BA$. Let's find the transpose of $BA$:
$(BA)^T = A^T B^T$
Since $A^T = -A$, substitute this into the expression:
$(BA)^T = (-A) B^T = -AB^T$
From the derived relation $AB^T = BA$, we can substitute $AB^T$ with $BA$:
$(BA)^T = -BA$
This condition means that $BA