Textbook Question
What expression in x represents ?
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Ch. 5 - Systems and Matrices
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 6, Problem 1
Answer each question. What is the product of and I2 (in either order)?
Verified step by step guidance1
Recall that the identity matrix \(I_2\) is a \(2 \times 2\) matrix given by \(I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\), which acts as the multiplicative identity for \(2 \times 2\) matrices.
Let the given \(2 \times 2\) matrix be \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\). The problem asks for the product of \(A\) and \(I_2\) in either order, so we need to find \(A \times I_2\) and \(I_2 \times A\).
To multiply \(A\) by \(I_2\), use the matrix multiplication rule: multiply each row of \(A\) by each column of \(I_2\). For example, the element in the first row and first column of the product is \(a \times 1 + b \times 0\).
Similarly, multiply \(I_2\) by \(A\) by taking each row of \(I_2\) and multiplying by each column of \(A\). For example, the element in the first row and first column of the product is \(1 \times a + 0 \times c\).
After performing the multiplications, observe that both \(A \times I_2\) and \(I_2 \times A\) result in the original matrix \(A\), confirming that \(I_2\) is the identity element for \(2 \times 2\) matrix multiplication.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Matrix Multiplication
Matrix multiplication involves combining two matrices by multiplying rows of the first matrix by columns of the second. The product is defined only when the number of columns in the first matrix equals the number of rows in the second. For square matrices, multiplication is always possible, but the order of multiplication can affect the result.
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Identity Matrix
The identity matrix, denoted I_n for an n×n matrix, is a square matrix with ones on the main diagonal and zeros elsewhere. It acts like the number 1 in matrix multiplication, meaning any matrix multiplied by the identity matrix of compatible size remains unchanged.
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Properties of Matrix Multiplication with Identity Matrix
Multiplying any square matrix by the identity matrix of the same size, in either order, results in the original matrix. This property confirms that the identity matrix is the multiplicative identity in matrix algebra, preserving the original matrix without alteration.
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Related Practice
Textbook Question
Answer each question. By what expression should we multiply each side of 5/((3x(2x + 1)) = A/(3x) + B/(2x + 1) so that there are no fractions in the equation?
Textbook Question
Answer each question. By what expression should we multiply each side of (3x - 2)/(x + 4)(3x^2 + 1) = A/(x + 4) + (Bx + C)/(3x^2 + 1) so that there are no fractions in the equation?
Textbook Question
What is the value of ?
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Textbook Question
Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (3,____).
x + y = 7
x2 + y2 = 25
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Textbook Question
Answer each question. What is the product ?
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