# MTH603 Quiz 3 Solution and Discussion

• A 3 x 3 identity matrix have three and different eigen values.
True
False

Eigenvalues and Eigenvectors of a 3 by 3 matrix. … The picture is more complicated, but as in the 2 by 2 case, our best insights come from finding the matrix’s eigenvectors: that is, those vectors whose direction the transformation leaves unchanged.

• Power method is applicable if the eigen vectors corresponding to eigen values are linearly independent.

Power method is applicable if the eigenvectors corresponding to eigenvalues are linearly independent.

True
False

linearly independent eigenvectors. … We could still compute the ratio of corresponding components for some index, but the … Using your power method code, try to determine the largest eigenvalue of the … So far, the methods we have discussed seem suitable for finding one or a few eigenvalues and eigenvectors at a time.

Reff

• The determinant of a diagonal matrix is the product of the diagonal elements.
True

The determinant of a lower triangular matrix (or an upper triangular matrix) is the product of the diagonal entries. In particular, the determinant of a diagonal matrix is the product of the diagonal entries.

Reff

• If we wanted to find the value of a definite integral with an infinite limit, we can instead replace the infinite limit with a variable, and then take the limit as this variable goes to _________.

constant
finite
infinity
zero

We will replace the infinity with a variable (usually t ), do the integral and then take the limit of the result as t goes to infinity. On a side note, notice that the area under a curve on an infinite interval was not infinity as we might have suspected it to be. In fact, it was a surprisingly small number.

• Euler’s Method numerically computes the approximate derivative of a function.

False

What is Euler method used for?
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

Reff

• An improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or 8 or -8 or, in some cases, as both endpoints approach limits.

True

• The Trapezoidal Rule is an improvement over using rectangles because we have much less “missing” from our calculations. We used ________ to model the curve in trapezoidal Rule.

• straight lines
• curves
• parabolas
• constant

Trapezoidal Rule, we used straight lines to model a curve and learned that it was an improvement over using rectangles for finding areas under curves because we had much less “missing” from each segment.

Reff

• In Runge – Kutta Method, we do not need to calculate higher order derivatives and find greater accuracy.

True

• The need of numerical integration arises for evaluating the definite integral of a function that has no explicit ____________ or whose antiderivative is not easy to obtain.

antiderivative
Or
derivative

The need often arises for evaluating the definite integral of a function that has no explicit antiderivative or whose antiderivative is not easy to obtain.

• An indefinite integral may _________ in the sense that the limit defining it may not exist.

An improper integral may diverge in the sense that the limit defining it may not exist. In this case, there are more sophisticated definitions of the limit which can produce a convergent value for the improper integral. These are called summability methods.

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