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Chapter 7: Problem 10

Use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{rr}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{rr}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{rr}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{rr}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{rr}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) \(C+F\)

Short Answer

Expert verified
C + F = \( \begin{bmatrix} 1 & 14 \\ 86 & 109 \\ 27 & 10 \end{bmatrix} \)

Step by step solution

01

Check Dimensions

To add two matrices, they must have the same dimensions. First, check the dimensions of matrices \(C\) and \(F\). Matrix \(C\) is 3x2 (3 rows and 2 columns) and matrix \(F\) is also 3x2 (3 rows and 2 columns). Since their dimensions match, we can proceed with the addition.
02

Add Corresponding Elements

Add the corresponding elements from matrices \(C\) and \(F\):- First row, first column: \(1 + 0 = 1\)- First row, second column: \(5 + 9 = 14\)- Second row, first column: \(8 + 78 = 86\)- Second row, second column: \(92 + 17 = 109\)- Third row, first column: \(12 + 15 = 27\)- Third row, second column: \(6 + 4 = 10\)Place these sums into a new matrix.
03

Write and Verify the Resulting Matrix

The resulting matrix from adding matrices \(C\) and \(F\) is:\[C+F = \begin{bmatrix} 1 & 14 \ 86 & 109 \ 27 & 10 \end{bmatrix}\]Check the work for accuracy by revisiting each calculation and confirming that each corresponding element was added correctly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Addition
Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices by combining their elements.
To perform matrix addition, the two matrices must have the same dimensions. This means that they should have the same number of rows and columns. If this condition is met, we can proceed to add the corresponding elements of each matrix.
For instance, if we have two matrices, matrix A and matrix B, with dimensions 2x2, their elements are added as follows:
  • Add the element in the first row and first column of matrix A to the element in the first row and first column of matrix B.
  • Continue this process for each corresponding element across the rows and columns.
Matrix addition is useful when dealing with systems of equations or transformations, as it allows for the combination of components in a straightforward manner.
Dimension Check
Before adding two matrices, it's crucial to verify that they have the same dimensions. The dimension of a matrix is described by the number of rows and columns it contains.
For example:
  • A matrix is described as "3x2" if it has 3 rows and 2 columns.
  • Another matrix of "3x2" dimensions can be added to the first one.
When dimensions match, you can proceed with matrix addition.
However, if they do not match, the operation is undefined, meaning it cannot be performed.
Checking dimensions is the first step in any matrix operation to ensure that the procedure is possible and to avoid calculation errors later on.
Corresponding Elements
The heart of matrix addition lies in the concept of corresponding elements. When matrices with identical dimensions are added, each element in the resultant matrix is formed by summing the corresponding elements from the two matrices.
To understand this better:
  • Consider two matrices, each of dimension 2x3. The element in the first row, first column of the resulting matrix is derived from the addition of the first row, first column elements of the two matrices.
  • This process is repeated for each element in the matrices.
Every position in the new matrix directly corresponds to a position in the original matrices.
By carefully matching these positions, you ensure that the process is accurate and that each element has been correctly computed.
Revising each calculation helps affirm the integrity of your resulting matrix.

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Most popular questions from this chapter

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer鈥檚 Rule. Three bands performed at a concert venue. The first band charged \(15 per ticket, the second band charged \)45 per ticket, and the final band charged \(22 per ticket. There were 510 tickets sold, for a total of \)12,700. If the first band had 40 more audience members than the second band, how many tickets were sold for each band?

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer鈥檚 Rule. At a women鈥檚 prison down the road, the total number of inmates aged 20鈥�49 totaled 5,525. This year, the 20鈥�29 age group increased by 10%, the 30鈥�39 age group decreased by 20%, and the 40鈥�49 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30鈥�39 age group than the 20鈥�29 age group. Determine the prison population for each age group last year.

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. At a competing cupcake store, \(\$ 4,520\) worth of cupcakes are sold daily. The chocolate cupcakes cost \(\$ 2.25\) and the red velvet cupcakes cost \(\$ 1.75 .\) If the total number of cupcakes sold per day is \(2,200,\) how many of each flavor are sold each day?

For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate- covered cashews. The nutritional information for these items is shown in Table 1 . $$ \begin{array}{|c|c|c|c|} \hline & \text { Fat (g) } & \text { Protein (g) } & \text { Carbohydrates (g) } \\ \hline \text { Almonds (10) } & 6 & 2 & 3 \\ \hline \text { Cranberries (10) } & 0.02 & 0 & 8 \\ \hline \text { Cashews (10) } & 7 & 3.5 & 5.5 \\ \hline \end{array} $$ For the "energy-booster" mix, there are 1,000 pieces in the mix, containing \(145 \mathrm{~g}\) of protein and \(625 \mathrm{~g}\) of carbohydrates. If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. \(30 \%\) of the almonds, \(20 \%\) of the cashews, and \(10 \%\) of the pistachios were eaten, and now there are 770 nuts left in he bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.
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