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Example Questions
Example Question #61 : Matrix Matrix Product
If , then which expression is equal toÂ
?
The transpose of a column matrix is the row matrix with the same entries, so
, soÂ
Find  by simply adding the products of corresponding entries:
SoÂ
Apply the trigonometric identity
,Â
setting :
Example Question #61 : Matrices
 is aÂ
 matrix.Â
 is aÂ
 matrix.Â
 andÂ
 are both nonsingular.
Which expression is defined?
 andÂ
, being nonsquare matrices, cannot have inverses, soÂ
 can be eliminated as a choice.
The product of a  matrix and aÂ
 matrix is aÂ
 matrix. Therefore:
 is aÂ
 matrix;Â
 andÂ
 are alsoÂ
 matrices.
Similarly,Â
,Â
, andÂ
 areÂ
 matrices.
Matrices of different dimensions cannot be added, so  ,Â
, andÂ
 can be eliminated.
 andÂ
 are Â
 andÂ
 matrices, respectively. Therefore,Â
 is aÂ
 matrix;Â
, the sum ofÂ
 matrices, is a defined expression.
Example Question #63 : Matrix Matrix Product
If , then which expression is equal toÂ
?
The transpose of a column matrix is the row matrix with the same entries, so if
,
then
Find  by simply adding the products of corresponding entries:
Apply the trigonometric identity
Setting ,
.
Example Question #62 : Matrices
Find .
None of the other choices gives the correct response.
None of the other choices gives the correct response.
 is a diagonal matrix. The product of two diagonal matrices can be found by multiplying elements in corresponding diagonal positions; the idea can be extended to powers of matrices, soÂ
By DeMoivre's Theorem,
Set  andÂ
 in the upper left element:
Set  andÂ
 in the lower right element:
Therefore,
,
which is not among the choices.
Example Question #751 : Linear Algebra
Which of the following is equal to ?
None of the other choices gives the correct response.
An easy way to find this is to note that ; therefore, we can findÂ
 by squaringÂ
 and squaring the result.Â
Matrix multiplication is worked row by column - each row in the former matrix is multiplied by each column in the latter by adding the products of elements in corresponding positions, as follows:
Now square this:
Example Question #63 : Matrices
Evaluate .
None of the other choices gives the correct response.
 is a diagonal matrix, so it can be raised to a power by raising the individual entries to that power:
By DeMoivre's Theorem,Â
.
We will need to rewrite the entries in the matrix as sums, not differences. We can do this by noting that the cosine and sine functions are even and odd, respectively, so  can be rewritten as
Applying DeMoivre's Theorem,
The coterminal angle for both  andÂ
 isÂ
, so
Â
Example Question #753 : Linear Algebra
 is a column matrix with seven entries. Which of the following is true ofÂ
 ?
 is a scalar.
 is a column matrix with seven entries.
 is a row matrix with seven entries.
 is aÂ
 matrix.
 is a matrix with a single entry.
 is a matrix with a single entry.
, a column matrix with ten entries, is aÂ
 matrix;Â
, its transpose, is aÂ
 matrix.Â
If  andÂ
 areÂ
 andÂ
 matrices, respectively, then the productÂ
 is aÂ
 matrix. Therefore,Â
 is aÂ
 matrix - a matrix with a single entry.
Example Question #754 : Linear Algebra
 is a column matrix with ten entries. Which of the following is true ofÂ
 ?
 is a column matrix with ten elements.
 is a matrix with a single entry.
 is a row matrix with ten elements.
 is aÂ
 matrix.
 is a scalar.
 is aÂ
 matrix.
, a column matrix with ten entries, is aÂ
 matrix;Â
, its transpose, is aÂ
 matrix.Â
If  andÂ
 areÂ
 andÂ
 matrices, respectively, then the productÂ
 is aÂ
 matrix. Therefore,Â
 is aÂ
 matrix.
Example Question #755 : Linear Algebra
Which of the following is equal to  ?
 andÂ
 are both elementary matrices, in that each can be formed from the (four-by-four) identity matrixÂ
 by a single row operation. Â
Since  differs fromÂ
 in that the entryÂ
 is in Row 2, Column 2, the row operation isÂ
. SinceÂ
 differs fromÂ
 in that entryÂ
 is in Row 3, Column 2, the row operation isÂ
.
Premultiplying a matrix by an elementary matrix has the effect of performing that row operation on the matrix. Looking at  asÂ
:
Premultiply  byÂ
 by performing the operationÂ
:
Premultiply  byÂ
 by performing the operationÂ
:
This is the correct product.
Example Question #756 : Linear Algebra
 refers to theÂ
 identity matrix.
. Which of the following matrices could be equal toÂ
 ?
None of the other responses gives a correct answer.
All of the matrices are diagonal, so the seventh power of each can be determined by simply taking the seventh power of the individual entries in the main diagonal. Also, note that each entry in each choice is of the formÂ
.
By DeMoivre's Theorem, for any real ,
Combining these ideas, we can take the seventh power of each matrix and determine which exponentiation yields the identity.
IfÂ
,
thenÂ
Â
IfÂ
,
thenÂ
Â
IfÂ
,
thenÂ
Â
IfÂ
,
thenÂ
Â
 is the only possible matrix value ofÂ
 among the choices.
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