Answer each question. What is the product of [ 6 4 − 1 8 ] \left[ \begin{matrix} 6 & 4 \\ -1 & 8 \end{matrix} \right] and I 2 (in either order)?

Hello, today we're going to be performing the multiplication of the given matrices. So what we are going to be multiplying is the matrix C three negative 27 and nine. And we're gonna be multiplying it with 100 and one. Now there's one thing to note about this matrix multiplication. The second matrix given to us is the identity matrix. For a two by two matrix, we have a property for multiplying anything by an identity matrix. Let's suppose that we have a as a two by two matrix. If we were to multiply any two by two matrix by a two by two identity matrix, then the resulting multiplication is just going to be the original two by two matrix. So what this means is that any two by two multiplied by the identity is just going to be itself. And that's exactly what we have here with this multiplication. We have our given two by two multiplied by the identity matrix of a two by two. And that means that the resulting multiplication is going to give us three negative 27 and nine. And this is going to be the product of the two given matrices. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

1:36 minutes

Problem 1

Textbook Question

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

Verified step by step guidance

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.

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.

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