The Transpose - Linear Algebra

To find the transpose, we need to make columns into rows.

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: ie. column 1 become row 1, column 2 becomes row 2, etc.

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: ie. column 1 become row 1, column 2 becomes row 2, etc.

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: ie. column 1 become row 1, column 2 becomes row 2, etc.

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: ie. column 1 become row 1, column 2 becomes row 2, etc.

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: ie. column 1 become row 1, column 2 becomes row 2, etc.

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: ie. column 1 become row 1, column 2 becomes row 2, etc.

The transverse is the matrix where the columns are now the corresponding rows - the first column is now the first row, the second column is now the second row, etc.

For a 3x2 matrix A, the transpose of A is a 2x3 matrix, where the columns are formed from the corresponding rows of A.

For a 2x4 matrix A, the transpose of A is a 4x2 matrix, where the columns are formed from the corresponding rows of A.

A square matrix is nonsingular - that is, it has an inverse - if and only if its determinant is 0.

Let a column matrix. Then the transpose is equal to the row matrix

.

The product of a row matrix and a column matrix, each with the same number of elements, is a one-by-one matrix whose only element is the sum of the squares of the elements, so

The determinant of a one-by-one matrix is its only entry, so is a square matrix with a nonzero determinant. This proves that can be nonsingular for some non-square .

Now, let , the two-by-two (or any other) identity matrix. is a nonsingular square matrix, and , so

.

This proves that can be nonsingular for some square .

Consequently, if , it cannot be determined whether or not is a nonsingular square matrix.