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Chapter 10: Problem 19

Solve the matrix equation for the unknown matrix \(X\), or explain why no solution exists. $$\begin{aligned} &A=\left[\begin{array}{ll} 4 & 6 \\ 1 & 3 \end{array}\right] \quad B=\left[\begin{array}{ll} 2 & 5 \\ 3 & 7 \end{array}\right]\\\ &C=\left[\begin{array}{ll} 2 & 3 \\ 1 & 0 \\ 0 & 2 \end{array}\right] \quad D=\left[\begin{array}{ll} 10 & 20 \\ 30 & 20 \\ 10 & 0 \end{array}\right] \end{aligned}$$ $$\frac{1}{5}(X+D)=C$$

Short Answer

Expert verified
The solution for \(X\) is matrix \[X = \begin{bmatrix} 0 & -5 \\ -25 & -20 \\ -10 & 10 \end{bmatrix} \].

Step by step solution

01

Understand the Matrix Equation

The given matrix equation is \( \frac{1}{5}(X + D) = C \). To find matrix \(X\), we first need to solve for \(X + D\) and then isolate \(X\).
02

Remove Scalar from Equation

Multiply both sides of the equation by 5 to eliminate the fraction: \(X + D = 5C\).
03

Calculate 5C

Multiply each element of matrix \(C\) by 5 to find \(5C\): \[ 5C = \begin{bmatrix} 5 \times 2 & 5 \times 3 \ 5 \times 1 & 5 \times 0 \ 5 \times 0 & 5 \times 2 \end{bmatrix} = \begin{bmatrix} 10 & 15 \ 5 & 0 \ 0 & 10 \end{bmatrix} \]
04

Set Up Equation for X

The equation \(X + D = 5C\) can be rewritten using matrix subtraction: \(X = 5C - D\).
05

Calculate X

Subtract matrix \(D\) from \(5C\): \[5C = \begin{bmatrix} 10 & 15 \ 5 & 0 \ 0 & 10 \end{bmatrix}, D = \begin{bmatrix} 10 & 20 \ 30 & 20 \ 10 & 0 \end{bmatrix} \]Subtract to find \(X\):\[ X = \begin{bmatrix} 10 - 10 & 15 - 20 \ 5 - 30 & 0 - 20 \ 0 - 10 & 10 - 0 \end{bmatrix} = \begin{bmatrix} 0 & -5 \ -25 & -20 \ -10 & 10 \end{bmatrix} \]
06

Verify Dimensions

Verify that the dimensions of \(X\) are consistent with the operations performed. Both \(5C\) and \(D\) are 3x2 matrices, hence their difference \(X\) is also a 3x2 matrix.

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Key Concepts

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

Matrix Multiplication
Matrix multiplication is a way to combine two matrices to form a new one. Not any two matrices can be multiplied. To do so, the number of columns in the first matrix must match the number of rows in the second matrix.
When multiplying matrices, you perform a series of dot products. A dot product is where you multiply corresponding elements from a row of the first matrix by elements from a column of the second matrix, and then add them all up. This yields a single number that forms part of a new matrix.
  • Matrix \(A\) is a 2x2 matrix, meaning it has 2 rows and 2 columns.
  • Matrix \(B\) is also a 2x2 matrix, therefore \(A\) and \(B\) can be multiplied together.
  • For each element in the resulting matrix, find the dot product of the corresponding row from \(A\) and column from \(B\).
Matrix multiplication is different from element-wise multiplication, where you simply multiply corresponding elements of each matrix without any summing.
Matrix Subtraction
Matrix subtraction is much like regular subtraction, but it is done element-wise between matrices. For matrix subtraction to be valid, the matrices involved must have the same dimensions.
Let鈥檚 take a look at the formula from the exercise: \(X = 5C - D\). This indicates that for each element in matrix \(X\), you will subtract the corresponding element from \(D\) from the element in \(5C\).
  • Subtract the top-left element of \(D\) from the top-left element of \(5C\).
  • Continue the same process for each corresponding element in the matrices.
  • Both matrices \(5C\) and \(D\) have dimensions 3x2, allowing matrix subtraction.
After subtracting, you get a new matrix with the same dimensions as the matrices you subtracted.
Matrix Dimensions
Understanding matrix dimensions is crucial for performing operations like multiplication and subtraction. The size of a matrix is described by its dimensions, given in rows and columns (e.g., 3x2 is 3 rows and 2 columns).
For the matrix equations in the exercise:
  • Matrix \(C\) has dimensions of 3x2.
  • Matrix \(D\) also has dimensions of 3x2.
  • Matrix \(X\), the solution to the equation, must also be 3x2 for the operations to be valid.
Matching dimensions are especially important when performing operations like addition or subtraction, as both matrices must be equally sized.
In the context of multiplication, the resulting matrix has dimensions defined by the outer dimensions of the matrices being multiplied.
Scalar Multiplication
Scalar multiplication involves multiplying each entry of a matrix by a constant, or "scalar". It's a straightforward process, yet fundamental.
For example, in the step where \(5C\) is calculated, each element of the original matrix \(C\) is multiplied by the scalar 5:
  • Multiply 2 by 5 to get 10.
  • Continue doing so for every element in matrix \(C\).
This operation scales the entire matrix, effectively multiplying every value inside it by the scalar. The resulting matrix maintains the same dimensions as the original.
Scalar multiplication is especially useful in many applications including scaling equations, stretching or shrinking geometric representations, and in preparations for further matrix operations.

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Most popular questions from this chapter

Square Roots of Matrices A square root of a matrix \(B\) is a matrix \(A\) with the property that \(A^{2}=B\). (This is the same definition as for a square root of a number.) Find as many square roots as you can of each matrix: $$\left[\begin{array}{ll} 4 & 0 \\ 0 & 9 \end{array}\right] \quad\left[\begin{array}{ll} 1 & 5 \\ 0 & 9 \end{array}\right]$$ [Hint: If \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) write the equations that \(a, b, c,\) and \(d\) would have to satisfy if \(A\) is the square root of the given matrix.]

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l} y \leq-2 x+8 \\ y \leq-\frac{1}{2} x+5 \\ x \geq 0, \quad y \geq 0 \end{array}\right.$$

Manufacturing Furniture A furniture factory makes wooden tables, chairs, and armoires. Each piece of furniture requires three operations: cutting the wood, assembling, and finishing. Each operation requires the number of hours ( \(h\) ) given in the table. The workers in the factory can provide 300 hours of cutting, 400 hours of assembling, and 590 hours of finishing each work week. How many tables, chairs, and armoires should be produced so that all available labor-hours are used? Or is this impossible? $$\begin{array}{|l|c|c|c|} \hline & \text { Table } & \text { Chair } & \text { Armoire } \\ \hline \text { Cutting (h) } & \frac{1}{2} & 1 & 1 \\ \text { Assembling (h) } & \frac{1}{2} & 1 \frac{1}{2} & 1 \\ \text { Finishing (h) } & 1 & 1 \frac{1}{2} & 2 \\ \hline \end{array}$$

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} x-6 y &=3 \\ 3 x+2 y &=1 \end{aligned}\right.$$

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} y & \geq x^{3} \\ y & \leq 2 x+4 \\ x+y & \geq 0 \end{aligned}\right.$$
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