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Example Questions
Example Question #91 : Linear Algebra
True or false:
is an example of a diagonal matrix.
True
False
True
A matrix is diagonal if and only if - that is, the element in column , row is equal to zero - for all . The given matrix fits this criterion.
Example Question #92 : Linear Algebra
True or false:
is an example of a diagonal matrix.
True
False
True
A matrix is diagonal if and only if - that is, the element in column , row is equal to zero - for all . The given matrix satisfies this condition, since its only nonzero elements are the first element in Column 1, the second element in Column 2, and so forth.
Example Question #15 : The Identity Matrix And Diagonal Matrices
Which of the following is equal to ?
is undefined.
is a diagonal matrix - its only nonzero entries are in its main diagonal, which comprises the elements in row , column , for .
The inverse of a such a matrix can be found simply by replacing each element in the main diagonal with its reciprocal. Rewriting the elements in the diagonal matrix as fractions,
,
or
.
Replace each diagonal element with its reciprocal to find :
Example Question #93 : Linear Algebra
True or false: is an example of a diagonal matrix.
True
False
False
A diagonal matrix has only zeroes as entries off of its main (upper-left to lower-right) diagonal. has three nonzero entries off this diagonal (Row 1, column 2, for example), so it is not a diagonal matrix.
Example Question #17 : The Identity Matrix And Diagonal Matrices
is a strictly diagonally dominant matrix for what range of values of ?
An matrix is strictly diagonally dominant if, for each , it holds that on Row , the absolute value of the element in Column is greater than the sum of the absolute values of the other elements in that row. Therefore, for to be strictly diagonally dominant, the following must hold:
For Row 1:
or
For Row 2:
or
For Row 3: ,
or
For all three conditions to hold, it is necessary that . This is the correct choice.
Example Question #18 : The Identity Matrix And Diagonal Matrices
is a diagonal matrix such that , where refers to the identity.
can be one of how many matrices?
is a diagonal matrix, and its dimensions are , so, for some complex and (the problem did not specify that the entries were real),
.
To raise a diagonal matrix to a power, simply raise each nonzero element to that power. It holds that
,
and, consequently, .
Equivalently, both and must be a one-hundredth root of 2, of which there are 100. Therefore, the number of possible matrices for is .
Example Question #94 : Linear Algebra
What is the trace of the identity matrix?
None of the other answers
None of the other answers
This question does not make sense since there is no such thing as the identity matrix, and it is not possible to take the trace of a matrix that is not square. This question is mostly meant to test your ability to think critically when reading certain mathematics problems.
Example Question #95 : Linear Algebra
True or false; all diagonal matrices are diagonalizable.
False
True
True
A matrix is diagonalizable if it can be written as , where is any invertible matrix, and is any diagonal matrix. If is already a diagonal matrix, we can of course write it as . Hence any diagonal matrix is diagonalizable.
Example Question #94 : Linear Algebra
Which of the following is an identity matrix?
All of these are valid identitiy matricies.
An identity matrix is a square matrix in which all diagonal elements are 1 and all non-diagonal elements are 0. The only square matricies above are:
The former of these two matricies is the only one fitting the critera of diagonal elements (diagonal meaning from top left to bottom right) being 1 and non-diagonal elements being 0.
Example Question #22 : Operations And Properties
Which of the following is a diagonal matrix?
A diagonal matrix is a matrix in which all elements except diagonal elements are zero. A diagonal element of a matrix M is the element Mij for which i=j. Here, i refers to the number of columns and j is the number of rows. The only matrix that fufills this definition is:
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