Understanding Matrix Transpose Properties

We are asked to evaluate the correctness of three statements regarding the transpose of matrices A, B, and C, which are of the same order. The transpose of a matrix A is denoted by A'. Let's examine each statement based on the fundamental properties of matrix transposes.

Statement 1: (A + B + C)' = A' + B’ + C’

This statement concerns the transpose of a sum of matrices. A well-known property of matrix transpose states that the transpose of the sum of two matrices is the sum of their transposes, i.e., \((X + Y)' = X' + Y'\). This property can be extended to any finite number of matrices being added together, provided they are of the same order so that addition is defined.

Applying this property to three matrices A, B, and C:

\((A + B + C)' = ((A + B) + C)'\)

Using the property for two matrices:

\(((A + B) + C)' = (A + B)' + C'\)

Applying the property again for \((A + B)'\):

\((A + B)' + C' = (A' + B') + C'\)

By associativity of matrix addition:

\((A' + B') + C' = A' + B' + C'\)

Thus, \((A + B + C)' = A' + B' + C'\) is a correct property. Statement 1 is correct.

Statement 2: (AB)’ = A’B’

This statement concerns the transpose of a product of two matrices. A fundamental property of matrix transpose states that the transpose of the product of two matrices is the product of their transposes in reverse order, i.e., \((XY)' = Y'X'\). This is a crucial property to remember.

Applying this property to matrices A and B:

\((AB)' = B'A'\)

The statement claims \((AB)' = A'B'\). This is generally incorrect unless A and B satisfy specific conditions (like A and B commuting and being symmetric, or one of them being zero, etc.). For arbitrary matrices A and B, the correct formula is \((AB)' = B'A'\).

Thus, Statement 2 is incorrect.

Statement 3: (ABC)’ = C’B’A’

This statement concerns the transpose of a product of three matrices. We can derive this by applying the property for two matrices iteratively. Let D = AB. Then (ABC)' = (DC)'.

Using the property \((XY)' = Y'X'\) for D and C:

\((DC)' = C'D'\)

Substitute D back: D' = (AB)'. Using the property \((XY)' = Y'X'\) again for A and B:

\(D' = (AB)' = B'A'\)

Now substitute D' back into \(C'D'\):

\(C'D' = C'(B'A')\)

By associativity of matrix multiplication:

\(C'(B'A') = C'B'A'\)

Thus, \((ABC)' = C'B'A'\) is a correct property. Statement 3 is correct.

Summary of Analysis

• Statement 1: \((A + B + C)' = A' + B' + C'\) - Correct (Transpose of a sum property extended).
• Statement 2: \((AB)' = A'B'\) - Incorrect (Correct property is \((AB)' = B'A'\)).
• Statement 3: \((ABC)' = C'B'A'\) - Correct (Transpose of a product property extended).

Based on the analysis, statements 1 and 3 are correct, while statement 2 is incorrect.

Conclusion

The correct statements are 1 and 3 only.

Revision Table: Matrix Transpose Properties

Additional Information: What is a Matrix Transpose?

The transpose of a matrix A, denoted by A' or \(A^T\), is a new matrix created by swapping the rows and columns of the original matrix. If A is an \(m \times n\) matrix, then its transpose A' will be an \(n \times m\) matrix.

Specifically, if the element in the \(i\)-th row and \(j\)-th column of matrix A is \(a_{ij}\), then the element in the \(j\)-th row and \(i\)-th column of the transpose A' is \(a'_{ji}\), and \(a'_{ji} = a_{ij}\).

Example:

Let matrix A be:

\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \]

Matrix A is a \(2 \times 3\) matrix. Its transpose, A', will be a \(3 \times 2\) matrix:

\[ A' = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix} \]

The first row of A becomes the first column of A', the second row of A becomes the second column of A', and so on.