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)?

<p>1 × 2 (B)</p> 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?

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

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

<p>A ≠ B. (A)</p> Signup and view all the answers

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

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

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

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

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

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

What is the result of the matrix A − B?

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

What is the matrix −A?

<p>(−3, −4, −2, 3, 0, −2) (B)</p> Signup and view all the answers

What is the result of the scalar multiplication 3A?

<p>(9, 12, 6, −9, 0, 6) (C)</p> Signup and view all the answers

How many elements does the matrix B have?

<p>6 (B)</p> Signup and view all the answers

What type of matrix is A?

<p>2x3 (D)</p> Signup and view all the answers

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

<p>(−3, −6, 3) (D)</p> Signup and view all the answers

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

<p>Add corresponding elements from both matrices (A)</p> Signup and view all the answers

Which of the following matrices is upper triangular?

<p>1 0 1 0 1 0 0 0 0 (C), 1 0 0 0 0 0 0 0 0 (D)</p> Signup and view all the answers

Which of these matrices is diagonal?

<p>2 0 0 0 0 0 0 0 0 (A), 1 0 0 0 0 0 0 0 1 (C)</p> Signup and view all the answers

Which of the following statements about triangular matrices is true?

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

Identify the condition that makes a matrix NOT triangular.

<p>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)</p> Signup and view all the answers

Which of these matrices is neither diagonal nor triangular?

<p>1 1 0 0 2 0 1 1 1 (B)</p> Signup and view all the answers

Which matrix can be classified as simply upper triangular?

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

What defines a square matrix?

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

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

<p>At least one non-zero element, rest are zero. (A)</p> 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?

<p>40 (D)</p> Signup and view all the answers

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

<p>4 (C)</p> Signup and view all the answers

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

<p>15 (A)</p> Signup and view all the answers

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

<p>400 (C)</p> Signup and view all the answers

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

<p>Connection through an intermediary group (A)</p> Signup and view all the answers

Which of the following best defines a square matrix?

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

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

<p>50 (A)</p> 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?

<p>15 (D)</p> 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?

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

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

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

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

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

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

<p>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)</p> 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?

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

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

<p>The vectors involved must be in ℝ n (B)</p> 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?

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

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

<p>It generates the vector space represented by A(ℝ n) (D)</p> 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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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.

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