Linear Algebra : Operations and Properties
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Example Questions
Example Question #191 : Linear Algebra
where 
Which expression is equal to the trace of 
The trace of a matrix is equal to the sum of the elements in its main diagonal - the elements going from upper left to lower right. Therefore,
Example Question #192 : Linear Algebra
Give the trace of 
The trace of a matrix is equal to the sum of the elements in its main diagonal, so the trace of 
Example Question #21 : The Trace
True or false: 
False
True
False
We can demonstrate this statement to be false with the counterexample
This matrix fits the definition of a diagonal matrix in that it has zeroes everywhere except on its main (upper-left-to-lower-right) diagonal. Also, its trace - the sum of the elements on its main diagonal - is
A matrix is singular - that is, without an inverse - if and only if its determinant is equal to 0. The determinant of a diagonal matrix is equal to the product of its diagonal elements, so
Example Question #121 : Operations And Properties
All skew-symmetric matrices have a trace of
It depends on the skew-symmetric matrix given
For any skew-symmetric matrix 
Example Question #22 : The Trace
Give the trace of 
The trace of the matrix is the sum of the elements on its main (upper-left corner to lower-right corner) diagonal:
Example Question #23 : The Trace
True or false: It follows that 
True
False
False
The statement is false, as can be demonstrated using the counterexample
This matrix is diagonal, so it can be squared by squaring its individual elements:
The trace of a matrix is equal to the sum of the elements on its main diagonal, so
but
This gives us a matrix 
Example Question #24 : The Trace
True or false:
True
False
True
The trace of a square matrix is equal to the sum of its diagonal elements - the elements whose row number and column number are equal. Therefore, if 
and
By the addition rule for finite sums,
Also, addition of matrices is elementwise, so
Example Question #124 : Operations And Properties
The characteristic equation of a matrix 
Which statement must be true?
None of the other statements are necessarily true.
The sum of the eigenvalues of a matrix is equal to the trace of the matrix. If 
When expanded, the coefficient of its 
Example Question #21 : The Trace
True or false: The trace of 
True
False
True
The trace of a matrix is the sum of the elements along its main diagonal, which are shown below:
The trace of 
Example Question #128 : Operations And Properties
Give the trace of
The trace of a matrix is the sum of the elements along its main diagonal, which are shown below:
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