Understanding Matrix Operations and Equality

The question asks us to evaluate the correctness of three statements involving matrix operations with given matrices A, B, and C. To do this, we need to perform the indicated matrix multiplications and additions and then compare the results.

The given matrices are:

• Matrix A: \(A = [m \ n]\) (a 1x2 matrix)
• Matrix B: \(B = [-n \ -m]\) (a 1x2 matrix)
• Matrix C: \(C = \begin{bmatrix} m \\ -m\end{bmatrix}\) (a 2x1 matrix)

Evaluating Statement 1: CA = CB

First, let's calculate the product CA. Matrix C is 2x1 and Matrix A is 1x2. The resulting matrix will be 2x2.

\( CA = \begin{bmatrix} m \\ -m\end{bmatrix} [m \ n] = \begin{bmatrix} m \cdot m & m \cdot n \\ (-m) \cdot m & (-m) \cdot n\end{bmatrix} = \begin{bmatrix} m^2 & mn \\ -m^2 & -mn\end{bmatrix} \)

Next, let's calculate the product CB. Matrix C is 2x1 and Matrix B is 1x2. The resulting matrix will be 2x2.

\( CB = \begin{bmatrix} m \\ -m\end{bmatrix} [-n \ -m] = \begin{bmatrix} m \cdot (-n) & m \cdot (-m) \\ (-m) \cdot (-n) & (-m) \cdot (-m)\end{bmatrix} = \begin{bmatrix} -mn & -m^2 \\ mn & m^2\end{bmatrix} \)

Now, let's compare CA and CB:

\( CA = \begin{bmatrix} m^2 & mn \\ -m^2 & -mn\end{bmatrix} \)

\( CB = \begin{bmatrix} -mn & -m^2 \\ mn & m^2\end{bmatrix} \)

For CA to be equal to CB, all corresponding elements must be equal. For example, \(m^2\) must equal \(-mn\), and \(mn\) must equal \(-m^2\). This is generally not true for arbitrary values of m and n (unless m=0 or n=0 under specific conditions). Therefore, statement 1 is incorrect.

Evaluating Statement 2: AC = BC

First, let's calculate the product AC. Matrix A is 1x2 and Matrix C is 2x1. The resulting matrix will be 1x1.

\( AC = [m \ n] \begin{bmatrix} m \\ -m\end{bmatrix} = [m \cdot m + n \cdot (-m)] = [m^2 - mn] \)

Next, let's calculate the product BC. Matrix B is 1x2 and Matrix C is 2x1. The resulting matrix will be 1x1.

\( BC = [-n \ -m] \begin{bmatrix} m \\ -m\end{bmatrix} = [(-n) \cdot m + (-m) \cdot (-m)] = [-mn + m^2] \)

Now, let's compare AC and BC:

\( AC = [m^2 - mn] \)

\( BC = [-mn + m^2] \)

Since \(m^2 - mn\) is equal to \(-mn + m^2\), the matrices AC and BC are equal. Therefore, statement 2 is correct.

Evaluating Statement 3: C(A + B) = CA + CB

This statement involves the distributive property of matrix multiplication over matrix addition. The property states that for matrices C, A, and B, if the operations are defined, then \(C(A+B) = CA + CB\).

First, let's check if the operations are defined:

• Matrix A (1x2) and Matrix B (1x2) have the same dimensions, so A + B is defined and is a 1x2 matrix.
• Matrix C (2x1) and Matrix A (1x2) can be multiplied (CA).
• Matrix C (2x1) and Matrix B (1x2) can be multiplied (CB).
• CA (2x2) and CB (2x2) have the same dimensions, so CA + CB is defined.
• Matrix C (2x1) and (A+B) (1x2) can be multiplied (C(A+B)).

Since all operations are defined and compatible with the dimensions, the distributive property \(C(A + B) = CA + CB\) holds true for these matrices.

Let's also verify by calculation:

Calculate A + B:

\( A + B = [m \ n] + [-n \ -m] = [m + (-n) \ n + (-m)] = [m-n \ n-m] \)

Calculate C(A + B): Matrix C is 2x1 and (A+B) is 1x2. The result is 2x2.

\( C(A + B) = \begin{bmatrix} m \\ -m\end{bmatrix} [m-n \ n-m] = \begin{bmatrix} m(m-n) & m(n-m) \\ -m(m-n) & -m(n-m)\end{bmatrix} = \begin{bmatrix} m^2-mn & mn-m^2 \\ -m^2+mn & -mn+m^2\end{bmatrix} \)

Calculate CA + CB: We already found \(CA = \begin{bmatrix} m^2 & mn \\ -m^2 & -mn\end{bmatrix}\) and \(CB = \begin{bmatrix} -mn & -m^2 \\ mn & m^2\end{bmatrix}\). Summing them:

\( CA + CB = \begin{bmatrix} m^2 & mn \\ -m^2 & -mn\end{bmatrix} + \begin{bmatrix} -mn & -m^2 \\ mn & m^2\end{bmatrix} = \begin{bmatrix} m^2 + (-mn) & mn + (-m^2) \\ -m^2 + mn & -mn + m^2\end{bmatrix} = \begin{bmatrix} m^2-mn & mn-m^2 \\ -m^2+mn & -mn+m^2\end{bmatrix} \)

Comparing \(C(A + B)\) and \(CA + CB\), we see that they are equal. Therefore, statement 3 is correct.

Conclusion

Based on our evaluations:

• Statement 1 (CA = CB) is incorrect.
• Statement 2 (AC = BC) is correct.
• Statement 3 (C(A + B) = CA + CB) is correct.

Thus, the correct statements are 2 and 3.

Revision Table: Key Matrix Properties

Additional Information on Matrix Equality

Two matrices are considered equal if and only if they meet two conditions:

1. They must have the same dimensions (the same number of rows and the same number of columns).
2. Each element in the first matrix must be equal to the corresponding element in the second matrix. That is, for matrices P and Q of the same dimensions, \(P_{ij} = Q_{ij}\) for all rows \(i\) and columns \(j\).

In statement 1 (CA = CB), both CA and CB are 2x2 matrices, so the dimension condition is met. However, as shown in the calculation, their corresponding elements are not equal for arbitrary m and n, proving the statement incorrect in general.

In statement 2 (AC = BC), both AC and BC are 1x1 matrices, meeting the dimension condition. Our calculation showed that the single elements of these 1x1 matrices, \(m^2 - mn\) and \(-mn + m^2\), are indeed equal. Thus, the equality holds.

Statement 3 relies on the distributive property, which is a fundamental rule of matrix algebra when the dimensions allow for the operations. Our verification also confirmed this through direct calculation.