Square Matrices Quiz

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Questions and Answers

What condition must be met for two matrices A and B to be considered equal?

A and B must be square matrices.

A and B must have the same size and all corresponding entries must be equal. (correct)

A and B must contain only integer entries.

A and B must be symmetric matrices.

If two matrices A and B are both 2 × 2 matrices, what can be concluded if A = B?

One matrix must be a scalar multiple of the other.

A and B cannot be used in matrix multiplication.

All corresponding entries in A and B are equal. (correct)

There is at least one entry that differs between A and B.

In Example 2.1.5, what values of x result in A being equal to B?

x = 3 and x = -3

x = -1 and x = 1

x = 2 and x = 4

x = 2 or x = -2 (correct)

What is the size of matrix A given as A = (a 2x + y, b 4x + 3y)?

1 × 2 (B) Signup and view all the answers

Which of the following systems of equations must be solved in Example 2.1.6 to determine x and y?

2x + y = 1, 4x + 3y = 6 (B) Signup and view all the answers

What can be concluded if two matrices A and B are of different sizes?

A ≠ B. (A) Signup and view all the answers

In the context of vectors, what condition allows two differently sized matrices to be equal?

They must represent the same mathematical object in a different dimensional space. (B) Signup and view all the answers

What entries must be equal for matrices A = (a ij) and B = (b ij) to satisfy A = B?

All corresponding entries a_ij must equal b_ij for all i and j. (D) Signup and view all the answers

What is the result of the matrix addition A + B?

(−1, 7, 5, −9, 1, 2) (B) Signup and view all the answers

What is the result of the matrix A − B?

(7, −1, 3, 3, 0, 2) (A) Signup and view all the answers

What is the matrix −A?

(−3, −4, −2, 3, 0, −2) (B) Signup and view all the answers

What is the result of the scalar multiplication 3A?

(9, 12, 6, −9, 0, 6) (C) Signup and view all the answers

How many elements does the matrix B have?

6 (B) Signup and view all the answers

What type of matrix is A?

2x3 (D) Signup and view all the answers

What is the sum of the second row of matrix A and the second row of matrix B?

(−3, −6, 3) (D) Signup and view all the answers

Which of the following represents the correct operation sequence to compute A + B?

Add corresponding elements from both matrices (A) Signup and view all the answers

Which of the following matrices is upper triangular?

1 0 1 0 1 0 0 0 0 (C), 1 0 0 0 0 0 0 0 0 (D) Signup and view all the answers

Which of these matrices is diagonal?

2 0 0 0 0 0 0 0 0 (A), 1 0 0 0 0 0 0 0 1 (C) Signup and view all the answers

Which of the following statements about triangular matrices is true?

Triangular matrices must be square matrices. (B), All elements below the main diagonal are zero in a triangular matrix. (C) Signup and view all the answers

Identify the condition that makes a matrix NOT triangular.

Having at least one non-zero element in the lower portion of the matrix. (B), Having non-zero elements both above and below the diagonal. (D) Signup and view all the answers

Which of these matrices is neither diagonal nor triangular?

1 1 0 0 2 0 1 1 1 (B) Signup and view all the answers

Which matrix can be classified as simply upper triangular?

2 3 0 0 5 0 0 0 0 (A), 1 0 1 0 0 0 0 0 0 (B) Signup and view all the answers

What defines a square matrix?

A matrix with an equal number of rows and columns. (C) Signup and view all the answers

How are diagonal matrices categorized based on their non-zero elements?

At least one non-zero element, rest are zero. (A) Signup and view all the answers

What is the trace of the matrix A given by entries a 11, a 22 of the order 2 square matrix?

40 (D) Signup and view all the answers

How many total direct contacts are represented between groups 2 and 3 in the given matrix B?

4 (C) Signup and view all the answers

What is the total indirect contact number between groups 1 and 3 according to the provided context?

15 (A) Signup and view all the answers

What unit profits correspond to item P2 according to the provided data?

400 (C) Signup and view all the answers

Which option represents the correct interpretation of an indirect contact between individuals in groups 1 and 3?

Connection through an intermediary group (A) Signup and view all the answers

Which of the following best defines a square matrix?

A matrix with the same number of rows and columns (D) Signup and view all the answers

In the context of the product items mentioned, what is the total tax collected in the first day?

50 (A) Signup and view all the answers

How many indirect contacts are there between the second individual in group 1 and the fourth individual in group 3?

15 (D) Signup and view all the answers

What is the relationship established by Theorem 2.2.2 regarding the set of column vectors of matrix A?

A(ℝ n) is equal to span S (B), A(ℝ n) is a subset of span S (C) Signup and view all the answers

If vector b is in A(ℝ n), what conclusion can be drawn about vector X?

X exists such that b = AX (D) Signup and view all the answers

What can be inferred if vector b is in span S according to Theorem 1.4.3?

There exists a vector X in ℝ n such that b = AX (B) Signup and view all the answers

Which statement accurately reflects the implications of the relationship between A(ℝ n) and span S?

Not all vectors in span S must be represented in A(ℝ n) (B), A(ℝ n) and span S can coincide but not be equal (D) Signup and view all the answers

How can a vector b be expressed in terms of the column vectors a1, a2, ..., an, according to the provided information?

As a sum of multiple scalar multiples of the column vectors (A) Signup and view all the answers

What does it imply if b = AX belongs to A(ℝ n)?

The vectors involved must be in ℝ n (B) Signup and view all the answers

According to the proof provided, if A(ℝ n) is a subset of span S, which statement also holds true?

There can exist vectors in span S that are independent of those in A(ℝ n) (C) Signup and view all the answers

What role does the matrix A play in defining the relationship between A(ℝ n) and span S?

It generates the vector space represented by A(ℝ n) (D) Signup and view all the answers

Study Notes

Square Matrices

A square matrix is an n x n matrix where the number of rows equals the number of columns.
The main diagonal of a square matrix is the diagonal from top left to bottom right.
The entries on the main diagonal are a11, a22, ..., ann.
The trace of a square matrix is the sum of the entries on the main diagonal, denoted as tr(A).
The trace of a square matrix A is calculated by: tr(A) = Σ aii (i = 1 to n), which is equivalent to a11 + a22 + ... + ann.

Example:

In the matrix A = (15 21; 44 25), the main diagonal entries are 15 and 25.
The trace of matrix A is tr(A) = 15 + 25 = 40.

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Description

Test your understanding of square matrices, including their properties and calculations. This quiz covers definitions, main diagonal, and how to compute the trace of a square matrix. Gain confidence in your knowledge of linear algebra.