Answer each question. What is the product of [64−18]\left[ \begin{matrix} 6 & 4 \\ -1 & 8 \end{matrix} \right] and I2 (in either order)?

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 1

6장, 문제 1

Answer each question. What is the product of $[\begin{matrix} 6 & 4 \\ - 1 & 8 \end{matrix}]$ and I₂ (in either order)?

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

Answer each question. What is the product of [64−18]\(\left\)[ \(\begin{matrix}\) 6 & 4 \\ -1 & 8 \(\end{matrix}\) \(\right\)] and I2 (in either order)?