transpose of a rectangular matrix is a ____________.
“What is the transpose of a rectangular matrix?”
Answer:
The transpose of a rectangular matrix is another matrix where the rows are switched with the columns. This means that if you have a matrix A of size m \times n (where m is the number of rows and n is the number of columns), the transpose of A, denoted A^T, will be of size n \times m.
To visualize this process:
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Conversion of rows into columns: Each row of the original matrix becomes a column in the transposed matrix.
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Conversion of columns into rows: Each column of the original matrix becomes a row in the transposed matrix.
Example
Let’s consider a matrix A:
A = \begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
Matrix A is a 2 \times 3 matrix because it has 2 rows and 3 columns.
The transpose of matrix A, denoted A^T, is:
A^T = \begin{bmatrix}
1 & 4 \\
2 & 5 \\
3 & 6
\end{bmatrix}
Now, A^T is a 3 \times 2 matrix because it has 3 rows and 2 columns.
Properties of Transpose
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Double Transpose: For any matrix A, the transpose of the transpose brings you back to the original matrix: (A^T)^T = A.
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Additive Property: The transpose of the sum of two matrices equals the sum of their transposes. If A and B are matrices of the same size, then (A + B)^T = A^T + B^T.
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Scalar Multiplication: For any scalar c and matrix A, the transpose of cA is cA^T: (cA)^T = cA^T.
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Product of Matrices: If A is an m \times n matrix and B is an n \times p matrix, then (AB)^T = B^T A^T. The transpose of a product is the product of their transposes in reverse order.
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Symmetric Matrix: A square matrix is symmetric if it is equal to its transpose, A = A^T.
The transpose of a matrix is a fundamental concept in linear algebra and plays a crucial role in various applications, including solving systems of linear equations, transforming and rotating geometric figures, and more.