Perform each operation, if possible. [ 3 2 5 ] − [ 8 − 4 6 ] + [ 1 0 2 ] \left[\begin{matrix}3\\ 2\\ 5\end{matrix}\right]-\left[\begin{matrix}8\\ -4\\ 6\end{matrix}\right]+\left[\begin{matrix}1\\ 0\\ 2\end{matrix}\right]

Hello, everyone. We are asked to apply the given operation to the following matrices. We have 33 by one matrices. The first one is the column reads down 416. We're subtracting that from the matrix where the column reads nine negative 57. And we're adding that to the matrix where the column reads 803, we have four answer choices. All of which are three by one matrices recall that when we add or subtract matrices, they must be of the same dimensions before we can add or subtract. In this case, all three are three by one matrices, which means we can do these operations. We are gonna keep the corresponding elements together. So my top element in the column will be four minus the element from the second matrix which is nine plus the first element from the third matrix which is eight, same with my second row. I will have one from the first matrix minus negative five from the second matrix plus zero. From the third matrix, third row be six from the first matrix minus seven from the second matrix plus three. From the third matrix simplifying this the first row I have four minus nine which is negative five, negative five plus eight is a positive three. Second row one minus negative five is a positive six plus zero is still a positive six. And then third row six minus seven is negative one, negative one plus three is a positive two. So our solution matrix is a three by one matrix where the column reads 362 and that matches answer choice A have a nice day.

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

2:31 minutes

Problem 73

Textbook Question

Perform each operation, if possible.
$[\begin{matrix} 3 \\ 2 \\ 5 \end{matrix}] - [\begin{matrix} 8 \\ - 4 \\ 6 \end{matrix}] + [\begin{matrix} 1 \\ 0 \\ 2 \end{matrix}]$

Verified step by step guidance

Identify the dimensions of each matrix involved in the operation. Here, all matrices are 1x3, meaning they each have 1 row and 3 columns.

Recall that matrix addition and subtraction are only defined for matrices of the same dimensions. Since all matrices are 1x3, these operations are possible.

Perform the subtraction first: subtract corresponding elements of the first and second 1x3 matrices. If the first matrix is $A = [a_1, a_2, a_3]$ and the second is $B = [b_1, b_2, b_3]$, then $A - B = [a_1 - b_1, a_2 - b_2, a_3 - b_3]$.

Next, add the resulting matrix from the subtraction to the third 1x3 matrix. If the third matrix is $C = [c_1, c_2, c_3]$, then $(A - B) + C = [(a_1 - b_1) + c_1, (a_2 - b_2) + c_2, (a_3 - b_3) + c_3]$.

Write the final matrix as the result of the operation, which will also be a 1x3 matrix with each element calculated as above.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Dimensions and Compatibility

Matrix operations like addition and subtraction require the matrices to have the same dimensions. Here, all matrices are 1x3, meaning they each have one row and three columns, so they are compatible for addition and subtraction.

Matrix Addition and Subtraction

Matrix addition and subtraction are performed element-wise. For two matrices of the same size, you add or subtract corresponding elements to get the resulting matrix.

Order of Operations in Matrix Arithmetic

When performing multiple operations, follow the order of operations (left to right for addition and subtraction). First subtract the second matrix from the first, then add the third matrix to the result.

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