In the following question there are statements which are TRUE and statements which are FALSE. Choose all the statements which are FALSE. 1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution. 2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined. 3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order. 4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O. 5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.

Question
Matrices
asked 2021-01-04
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.

Answers (1)

2021-01-05
Step 1
Answer(1):
The given statement is:
If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
The given statement is false because it's not compulsory that, if If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent, it can be consistent.
For example:
\(\begin{cases} x=8\\2x=16 \end{cases}\)
In this system, the number of equations is two and the number of variables is one, that is the number of equations is exceeded by the number of variables.
Thus, this system is consistent.
Step 2
Answer(2):
The given statement is:
If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
The given statement is false because it's not compulsory that, If B has a column with zeros, then AB will also have a column with zeros if this product is defined.
For example:
Consider, \(A=\begin{bmatrix}1 & 0 \\2 & 1 \end{bmatrix} , B=\begin{bmatrix}3 & 0 \\2 & 0 \end{bmatrix}\) are two matrices, then the product of these matrices is given as:
\(AB=\begin{bmatrix}(1\times3+0\times2) & (2\times3+1\times2) \\(1\times0+0\times0) & (2\times0+1\times0) \end{bmatrix}\)
\(AB=\begin{bmatrix}3 & 8 \\0 & 0 \end{bmatrix}\)
Clearly, their product is defined, and not any column is zero.
Therefore, the given statement is false.
Step 3
Answer(4):
The given statement is:
Suppose A is an n x n matrix and assume \(A^2 = O\), where O is the zero matrices. Then A = O.
The given statement is false because it's not compulsory that, If A is an n x n matrix and assume \(A^2 = O\), where O is the zero matrices. Then A = O.
For example:
Consider, \(A=\begin{bmatrix}0 & 0 \\1 & 0 \end{bmatrix}\) is a non-zero matrix but its square is zero.
Therefore, the given statement is false.
Step 4
Hence, all false statements are 1, 2 and 4.
0

Relevant Questions

asked 2020-12-25
Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
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1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
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4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
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then AB=?
BA=?
True or false : AB=BA for any two square matrices A and B of the same size.
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