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Matrices Trivia Quiz

Matrices are used in the field of encryption and computer graphics. How much do you know about them? Enjoy and thanks for playing.

A multiple-choice quiz by Matthew_07. Estimated time: 5 mins.

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Author
Matthew_07

Time
5 mins

Type
Multiple Choice

Quiz #
274,481

Updated
Sep 24 23

# Qns
10

Difficulty
Tough

Avg Score
5 / 10

Plays
1078

One at a Time - Single Page - Timed Game

Question 1 of 10

1. Matrix operations possess many properties. Let A, B and C be 3 matrices of the same size n x n, and d and e be 2 different real number scalars. Which of the following is NOT one of the properties? Hint

d(AB) = (dA)B = A(dB)

(A+B)^2 = A^2 + 2AB + B^2

d(eA) = (de)A

A(BC) = (AB)C

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Question 2 of 10

2. The inverse of a matrix, if exists, is unique. The kind of matrix which has an inverse matrix is called? Hint

A nonsingular matrix

A coupling matrix

A singular matrix

A non-coupling matrix

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Question 3 of 10

3. The inverse of a matrix can be obtained by multiplying the reciprocal of the matrix's determinant with its? Hint

Minor

Cofactor

Transpose

Adjoint

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Question 4 of 10

4. If matrix A is invertible, then which of the followings is NOT true? (A^T is the transpose of A; a is any non-zero real number scalar; n is any positive integer) Hint

(aA)^-1 = (1/a)(A^-1)

(A^T)^-1 = (A^-1)^T

A = A^-1

(A^n)-1 = (A^-1)^n

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Question 5 of 10

5. The rank of a matrix A is denoted as rank (a), which is defined as? Hint

The number of nonzero columns in its row echelon form.

The number of nonzero rows in its row echelon form.

The number of zero columns in its row echelon form.

The number of zero rows in its row echelon form.

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Question 6 of 10

6. Let AX = B be a linear system of m equations and n unknowns. Not all systems have a solution. A system that does not have a solution is in the form of? [rank (A/B) means rank (A|B)] Hint

rank (A) = rank (A/B) = n

rank (A) < rank (A/B)

rank (A) = rank (A/B) < n

rank (A) > rank (A/B)

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Question 7 of 10

7. A square matrix A is called a symmetric matrix if its transpose is the same as the matrix itself, namely A^T = A. On the other hand, a skew-symmetrical matrix is the one that fits in which of the following equations? Hint

A^T = surd A

A^T = A^2

A^T = -A

A^T = 1/A

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Question 8 of 10

8. An elementary matrix can be obtained by performing any one of the three operations on an identity matrix. Which of the followings is NOT one of them? Hint

Interchanging 2 rows

Squaring all values in a row

Multiplying a row by a non-zero constant

Adding a multiple of 1 row to another row

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Question 9 of 10

9. A matrix's row echelon form is ______; while its reduced row echelon form is ______. Hint

Unique; unique

Not unique; not unique

Unique; not unique

Not unique; unique

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Question 10 of 10

10. We can "decompose" a matrix into 2 "simpler" matrices by using the LU-Decomposition method. This method comes in useful when solving the linear system of equations. LU as in LU-Decomposition stands for? Hint

Lower-Under

Long-Under

Lower-Upper

Long-Upper

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Apr 05 2024 : Guest 102: 5/10
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Quiz Answer Key and Fun Facts

Answer: (A+B)^2 = A^2 + 2AB + B^2

In general, (A+B)^2 is not equal to A^2 + 2AB + B^2, unless AB = BA. Actually, (A+B)^2 = (A+B)(A+B) = A^2 + AB + BA + B^2. The order of arrangement in matrix multiplication operation is very important, unlike the normal real number multiplication operation. We can write 6 = 1 x 6 = 6 x 1, but usually, AB is not equal to BA.

2. The inverse of a matrix, if exists, is unique. The kind of matrix which has an inverse matrix is called?

Answer: A nonsingular matrix

All elementary matrices are nonsingular, meaning that they all have their inverse pairs.

3. The inverse of a matrix can be obtained by multiplying the reciprocal of the matrix's determinant with its?

Answer: Adjoint

The formula is given by A^-1 = (1/|A|) x Adj (A), where A^-1 is the inverse matrix, |A| is the matrix's determinant, and Adj(A) is the adjoint of the matrix.

4. If matrix A is invertible, then which of the followings is NOT true? (A^T is the transpose of A; a is any non-zero real number scalar; n is any positive integer)

Answer: A = A^-1

In general, A is not equal to A^-1. Besides, it is interesting to note that (AB)^-1 = (B^-1)(A^-1), and (AB)^T = (B^T)(A^T).

5. The rank of a matrix A is denoted as rank (a), which is defined as?

Answer: The number of nonzero rows in its row echelon form.

The rank of a matrix, either in its row echelon form, or its reduced row echelon form, is the same.

6. Let AX = B be a linear system of m equations and n unknowns. Not all systems have a solution. A system that does not have a solution is in the form of? [rank (A/B) means rank (A|B)]

Answer: rank (A) < rank (A/B)

A system that has a unique solution is in the form of rank (A) equals to rank (A|B) equals to n. On the other hand, the one that has infinitely many solutions is in the form of rank (A) equals to rank (A|B) less than n. Notice that rank (A) for an augmented matrix (A|B) is always equals or smaller than rank (A|B). Hence, it is impossible to have the inequality of rank (A) > rank (A|B).

Answer: A^T = -A

For example, A + A^T is a symmetric matrix, because (A + A^T)^T = A^T + (A^T)^T = A^T + A = A + A^T.

8. An elementary matrix can be obtained by performing any one of the three operations on an identity matrix. Which of the followings is NOT one of them?

Answer: Squaring all values in a row

These 3 operations are called the elementary row operations (ERO).

9. A matrix's row echelon form is ______; while its reduced row echelon form is ______.

Answer: Not unique; unique

Some additional steps can be carried on to reduce a row echelon form of a matrix to its reduced row echelon form.

Answer: Lower-Upper

It means lower triangular and upper triangular matrices. By writing AX=B in LUX=B, we let Y = UX, we get LY = B. We can then easily find the values of elements in the matrix Y. Next, we solve for all the values in X using the equation Y = UX.

Source: Author Matthew_07

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