All Linear Algebra Resources
Example Questions
Example Question #41 : Matrices
Find the product .
,
cannot be multiplied
If is an matrix and is an matrix,
can only be multiplied if , or the number of columns in equals the number of rows in . Otherwise there is a mismatch, and the two matrices can not be multiplied. will be an matrix
Since is a matrix and is a matrix, then can be multiplied and will have the dimensions .
To find the product , you must find the dot product of the rows of and the columns of
,
We find by finding the dot product of the row of and column of .
We find by finding the dot product of the row of and column of .
We use the same method to find the rest of the matrix values
Example Question #42 : Matrices
Find the product .
,
cannot be multiplied.
cannot be multiplied.
If is an matrix and is an matrix,
can only be multiplied if , or the number of columns in equals the number of rows in . Otherwise there is a mismatch, and the two matrices can not be multiplied. will be an matrix
Since is a matrix and is a matrix, so cannot be multiplied.
Example Question #723 : Linear Algebra
Find the product .
,
cannot be multiplied
If is an matrix and is an matrix,
can only be multiplied if , or the number of columns in equals the number of rows in . Otherwise there is a mismatch, and the two matrices can not be multiplied. will be an matrix
Since is a matrix and is a matrix, then can be multiplied and will have the dimensions .
To find the product , you must find the dot product of the rows of and the columns of
,
We find by finding the dot product of the row of and column of .
We find by finding the dot product of the row of and column of .
We use the same method to find the rest of the matrix values
Example Question #724 : Linear Algebra
Find the product .
,
cannot be multiplied
If is an matrix and is an matrix,
can only be multiplied if , or the number of columns in equals the number of rows in . Otherwise there is a mismatch, and the two matrices can not be multiplied. will be an matrix
Since is a matrix and is a matrix, then can be multiplied and will have the dimensions .
To find the product , you must find the dot product of the rows of and the columns of
,
We find by finding the dot product of the row of and column of .
We find by finding the dot product of the row of and column of .
We use the same method to find the rest of the matrix values
Example Question #42 : Matrices
Find the product .
,
cannot be multiplied.
cannot be multiplied.
If is an matrix and is an matrix,
can only be multiplied if , or the number of columns in equals the number of rows in . Otherwise there is a mismatch, and the two matrices can not be multiplied. will be an matrix
Since is a matrix and is a matrix, so cannot be multiplied.
Example Question #731 : Linear Algebra
True or false:
is an example of an idempotent matrix.
False
True
True
is an idempotent matrix, by definition, if . Multiply by itself by multiplying rows by columns - multiplying elements in corresponding positions and adding the products:
.
, making idempotent.
Example Question #732 : Linear Algebra
True or false:
is an example of an idempotent matrix.
False
True
False
is an idempotent matrix, by definition, if . Multiply by itself by multiplying rows by columns - multiplying elements in corresponding positions and adding the products:
, so is not idempotent.
Example Question #733 : Linear Algebra
For any given value , how many nonsingular idempotent matrices exist?
Zero
Infinitely many
One
Two
One
is nonsingular, by definition, if it has an inverse - that is, if exists. is an idempotent matrix, by definition, if
Premultiplying both sides of the equation by , we get
,
where is the identity matrix.
Matrix multiplication is associative, so
.
Therefore, the only nonsingular idempotent matrix of a given dimension is the identity.
Example Question #734 : Linear Algebra
The above diagram shows a board for a game of chance. A player moves according to the flip of a fair coin, depending on his current location. For example, if he is on the green square, he will move to the orange square if the coin comes up heads, and to the pink square if it comes up tails.
Construct a stochastic matrix that models this game, with the rows/columns representing, in order, the orange, pink, blue, and green squares.
Since the coin is fair, each outcome will come up with probability 0.5. Therefore, a player on orange end up on orange or pink with 0.5 probability each; a player on pink will end up on orange or blue with probability 0.5 each; a player on blue will end up on pink or green with probability 0.5 each; and a player on green will end up on orange or pink with probability 0.5 each. The matrix that models this is
.
Example Question #735 : Linear Algebra
It is recommended that you use a calculator with matrix arithmetic capability for this problem.
The above diagram shows a board for a game of chance. A player moves according to the flip of a fair coin, depending on his current location. For example, if he is on the green square, he will move to the orange square if the coin comes up heads, and to the pink square if it comes up tails.
The player agrees to start on the pink square. Which square is he most likely to end up on after eight moves?
Orange
Pink
Green
Blue
Orange
Since the coin is fair, each outcome will come up with probability 0.5. Therefore, a player on orange end up on orange or pink with 0.5 probability each; a player on pink will end up on orange or blue with probability 0.5 each; a player on blue will end up on pink or green with probability 0.5 each; and a player on green will end up on orange or pink with probability 0.5 each. The matrix that models this is
.
The matrix that models the probabilities that the player will end up on a given square, given that he starts on a particular square, is , which through calculation is
Since the player is starting on pink, examine the second column; the greatest entry is in the second row, which represents ending up on orange.
Certified Tutor