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Example Questions
Example Question #21 : The Transpose
and
are both lower triangular square matrices. Which of the following must follow from this?
is a zero matrix
is an identity matrix
is a nonsingular matrix, but not necessarily the identity matrix.
is a singular matrix, but not necessarily a zero matrix.
is a diagonal matrix, but not necessarily an identity matrix or a zero matrix.
is a diagonal matrix, but not necessarily an identity matrix or a zero matrix.
Let be a three-by-three matrix; this reasoning extends to square matrices of all sizes.
is a lower triangular matrix, so all of the entries above its main (upper left to lower right) diagonal are zeroes; that is,
.
, the transpose of
, is the matrix formed by switching rows with columns, so
.
However, is lower triangular also; as a consequence,
,
and
.
This demonstrates that must be a diagonal matrix - one with only zeroes off its main diagonal.
Example Question #51 : Operations And Properties
Let ,
, and
be real numbers such that
,
,
and the determinant of is 8.
True or false: The determinant of is 8.
False
True
True
is the transpose of
- the matrix formed by interchanging the rows of
with its columns. The determinant of a matrix and that of its transpose are equal, so, since
has determinant 8, so does
.
Example Question #131 : Linear Algebra
Find .
, the conjugate transpose of
, is the result of transposing the matrix - interchanging rows with columns - and changing each entry to its complex conjugate. First, find transpose
:
,
so
Change each entry to its complex conjugate:
.
Example Question #132 : Linear Algebra
Find .
None of the other choices gives the correct response.
, the transpose, is the result of switching the rows of
with the columns.
,
so
.
Example Question #133 : Linear Algebra
and
are skew-symmetric matrices.
Which of the following is true of ?
By definition, the transpose of a skew-symmetric matrix
is equal to its additive inverse
. It follows that
Example Question #134 : Linear Algebra
Find .
, the conjugate transpose of
, is the result of transposing the matrix - interchanging rows with columns - and changing each entry to its complex conjugate. First, find transpose
:
,
so
Change each entry to its complex conjugate:
Example Question #56 : Operations And Properties
Which of the following is equal to ?
None of the other responses gives the correct answer.
, the conjugate transpose of
, is the result of transposing the matrix - interchanging rows with columns - and changing each entry to its complex conjugate. First, find transpose
:
Each entry of is equal to the complex conjugate of the corresponding entry of
. However, each entry in
is real, so each entry is equal to its own complex conjugate, and
Example Question #135 : Linear Algebra
Find .
None of the other choices gives the correct response.
, the transpose, is the result of switching the rows of
with the columns.
,
so
.
Example Question #31 : The Transpose
Which of the following is equal to ?
is the transpose of
- the result of interchanging the rows of
with its columns.
is the conjugate transpose of
- the result of changing each entry of
to its complex conjugate. Therefore, if
,
we can find by simply changing each entry in
to its complex conjugate:
Example Question #61 : Operations And Properties
True or false: is an upper triangular matrix.
True
False
True
is the result of interchanging rows of
with columns, then changing each entry to its complex conjugate. Also,
is equal to
, so perform the same process on
:
A matrix is upper triangular if all elements below its main (upper-left corner to lower right corner) diagonal are equal to 0. These elements in are displayed in red above. Since all of the lower-triangular elements of
are zeroes,
is upper triangular.
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