The Transpose - Linear Algebra
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Find the transpose of Matrix 
.
Find the transpose of Matrix
To find the transpose, we need to make columns into rows.
To find the transpose, we need to make columns into rows.
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Transpose matrix A where,
Transpose matrix A where,
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:
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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:
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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:
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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:
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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:
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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:
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Which is the transpose of 
?
Which is the transpose of
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.
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.
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Find the transpose of matrix A.
Find the transpose of matrix A.
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 3x2 matrix A, the transpose of A is a 2x3 matrix, where the columns are formed from the corresponding rows of A.
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Find the transpose of matrix A.
Find the transpose of matrix 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.
For a 2x4 matrix A, the transpose of A is a 4x2 matrix, where the columns are formed from the corresponding rows of A.
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is a nonsingular square matrix.
True, false, or undetermined: 
is a nonsingular square matrix.
True, false, or undetermined:
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.
A square matrix is nonsingular - that is, it has an inverse - if and only if its determinant is 0.
Let
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
Now, let
This proves that
Consequently, if
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