Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve for \(u, x, y\), and \(z\) in the given matrix equation. \(\left[\begin{array}{rr}1 & 2 \\ 3 & 4 \\ x & -1\end{array}\right]-3\left[\begin{array}{cc}y-1 & 2 \\ 1 & 2 \\ 4 & 2 z+1\end{array}\right]=2\left[\begin{array}{rr}-4 & -u \\ 0 & -1 \\ 4 & 4\end{array}\right]\)

Step-by-Step Solution

Answer

The values for the variables in the given matrix equation are \(u=-4\), \(x=-4\), \(y=4\), and \(z=1\).

Step 1: Distribute the Scalars

First, we need to multiply each matrix by its scalar. For the second and third matrices, that means distributing -3 and 2, respectively, to each element in the matrices. Matrix1 = $$ \left[\begin{array}{rr}1 & 2 \\\ 3 & 4 \\\ x & -1\end{array}\right] $$ Matrix2 = \(3\left[\begin{array}{rr}y-1 & 2 \\\ 1 & 2 \\\ 4 &2z+1\end{array}\right]\) which gives: Matrix2 = $$ \left[\begin{array}{rr}3(y-1) & -6 \\\ -3 & -6 \\\ -12 & -6(2z+1)\end{array}\right] $$ Matrix3 = \(2\left[\begin{array}{rr}-4 & -u \\\ 0 & -1 \\\ 4 & 4\end{array}\right]\) which gives: Matrix3 = $$ \left[\begin{array}{rr}-8 & -2u \\\ 0 & -2 \\\ 8 & 8\end{array}\right] $$

Step 2: Combine Matrices

Now we need to combine the matrices using the given equation: Matrix1 - Matrix2 = Matrix3 Let's compute the matrix subtraction and set each element equal to the corresponding element in Matrix3. $$ \left[\begin{array}{rr}1 - 3(y-1) & 2 + 6 \\\ 3 +3 & 4 + 6 \\\ x + 12 & -1 + 6(2z+1)\end{array}\right] = \left[\begin{array}{rr}-8 & -2u \\\ 0 & -2 \\\ 8 & 8\end{array}\right] $$ This results in the following equations: 1. \(1 -3(y-1) = -8\) 2. \(2+6 = -2u\) 3. \(3+3 = 0\) 4. \(4+6 = -2\) 5. \(x+12 = 8\) 6. \(-1+6(2z+1) = 8\)

Step 3: Solve for Variables

Now let's solve each equation for the corresponding variable: 1. \(1 - 3y + 3 = -8 \Rightarrow -3y+4 = -8 \Rightarrow y=4\) 2. \(8=-2u \Rightarrow u=-4\) 3. This equation is incorrect, so it will not help us to find any variable. 4. This equation is incorrect, so it will not help us to find any variable. 5. \(x+ 12 = 8 \Rightarrow x=-4\) 6. \(-1+6(2(4)+1) = 8 \Rightarrow -1+48 = 8 \Rightarrow z=1\) The values for the variables are \(u=-4\), \(x=-4\), \(y=4\), and \(z=1\).

Key Concepts

Matrix Operations

A matrix is a rectangular array of numbers arranged in rows and columns. Matrix operations involve various actions like addition, subtraction, multiplication, and more.
With matrix operations, it can be crucial to ensure the dimensions align, especially when performing addition and subtraction. If matrices do not have the same dimensions, these operations are not defined.
In this specific exercise, the matrix equation involves subtraction and scalar multiplication. Understanding matrix operations, especially how they correspond element-wise, is vital when solving such questions.

Scalar Multiplication

Scalar multiplication is a method where you multiply a matrix by a single number called a scalar. This process scales the entire matrix by multiplying every individual element by the scalar value.
For example, if we have a scalar of 3 and a matrix \(\left[\begin{array}{rr}y-1 & 2 \ 1 & 2 \ 4 & 2z+1\end{array}\right]\), scalar multiplication would result in \(3\left[\begin{array}{rr}y-1 & 2 \ 1 & 2 \ 4 & 2z+1\end{array}\right] = \left[\begin{array}{rr}3(y-1) & 6 \ 3 & 6 \ 12 & 3(2z+1)\end{array}\right]\).
Every element is multiplied by 3. Hence, understanding scalar multiplication is crucial when simplifying matrix equations during matrix operations.

Matrix Subtraction

Matrix subtraction is similar to matrix addition but involves subtracting corresponding elements of two matrices of the same size. It is an element-wise operation.
For example, in our exercise, we have Matrix1 and Matrix2:
\(\left[\begin{array}{rr}1 & 2 \ 3 & 4 \ x & -1\end{array}\right]\) and \(3\left[\begin{array}{rr}y-1 & 2 \ 1 & 2 \ 4 & 2z+1\end{array}\right]\).
When subtracting Matrix2 from Matrix1, you need to subtract the individual elements from one another. This results in another matrix, and comparing this resultant matrix with another will help solve for unknowns. Ensuring both matrices are the same size is essential for performing valid subtraction.

Solving Systems of Equations

Solving systems of equations involves finding values for variables that satisfy multiple equations simultaneously. In the context of matrices, solving these systems often involves equating elements from both sides of an equation after performing operations like addition, subtraction, or scalar multiplication on matrices.
After performing such operations in the exercise, we derived a set of equations from the matrix equality, such as \(1 - 3(y-1) = -8\) or \(x + 12 = 8\). By solving these equations individually, you can find the values of unknowns like \(u, x, y\), and \(z\).
Understanding how to tackle the unknowns systematically is essential for solving matrix-based problems efficiently. Approaching them with strategies like back-substitution can often help simplify finding solutions.