Problem 23 - Algebra and Trigonometry

Question

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ B-X=4 A $$

Step-by-Step Solution

Answer

The solution matrix X is \[ X = \left[\begin{array}{cc} 7 & 27 \ -8 & 36 \ -17 & -4 \end{array}\right] \]

Step 1: Understand the Matrix Equation

Analyze the matrix equation. The given equation is B - X = 4A. The goal is to find the matrix X. Understand that '4A' means each entry of matrix A is scaled by a factor of 4.

Step 2: Compute 4A

Compute the matrix 4A by multiplying each element of matrix A by 4. \[ 4A = \left[\begin{array}{cc} -12 & -28 \ 8 & -36 \ 20 & 0 \end{array}\right] \]

Step 3: Solve for X

Now, since B - X = 4A, we can find X by rearranging the equation like B - 4A = X. Let's substitute B and 4A into this equation.\[ X = \left[\begin{array}{cc} -5 & -1 \ 0 & 0 \ 3 & -4 \end{array}\right] - \left[\begin{array}{cc} -12 & -28 \ 8 & -36 \ 20 & 0 \end{array}\right]= \left[\begin{array}{cc} 7 & 27 \ -8 & 36 \ -17 & -4 \end{array}\right] \]

Key Concepts

Matrices Algebra

Matrices algebra involves operations that you can perform on matrices – which are grid-like arrays of numbers. Understanding matrices is crucial because they are a cornerstone in various areas of mathematics and its applications. In our exercise, the process of finding matrix X when given matrices A and B and the expression \( B-X=4A \) demonstrates one aspect of matrices algebra, namely the combination of scalar multiplication and matrix subtraction.

Scalar multiplication, in this context, involves multiplying a scalar (a single number) by each entry of a matrix. You can think of it as inflating or deflating the values of the matrix uniformly across all entries. For example, multiplying matrix A by 4, the scalar, scales up every number in A by four times. Meanwhile, subtraction in matrices algebra means taking two matrices of the same size and subtracting their corresponding entries to produce a new matrix. As seen in the solution, this is how we isolate X to solve the given matrix equation.

Matrix Subtraction

In the realm of matrices, subtraction might seem straightforward, but it requires attention to detail. Matrix subtraction can only occur between two matrices of the same dimensions. That is, they must have the same number of rows and columns. The process is similar to addition; you subtract corresponding elements from the matrices involved. For instance, if we have two matrices C and D, where \( C_{ij} \) and \( D_{ij} \) represent their elements at row i and column j, the resulting matrix E after the operation \( C - D \) will have elements \( E_{ij} = C_{ij} - D_{ij} \).

Our exercise demonstrates this with matrices B and 4A, which are both 3 by 2 matrices. They subtract element-wise to yield matrix X. It's a process that requires careful alignment of corresponding elements, ensuring that the subtraction of numbers is done correctly. The calculate-by-hand approach is systematic, subtracting one pair of numbers at a time to ensure accuracy.

Scalar Multiplication of Matrices

Scalar multiplication is the operation where each entry in a matrix is multiplied by a scalar (a real number). This differs from matrix multiplication in that every element of the matrix is affected uniformly, and the dimension of the matrix remains unchanged. In the exercise, the scalar multiplication involves multiplying every element of matrix A by 4. This action augments the value of each element, effectively resizing the matrix values while keeping the structure identical.

In practice, scalar multiplication is performed by taking the scalar and individually multiplying it across every entry of the matrix. When instructing on such operations, ensure students understand the uniformity associated with this process – the same scalar applied to every single entry without exception. Visual aids or color coding of matrix entries can be particularly helpful in demonstrating the scalar's distribution across all elements in a matrix.