Problem 20 - College Algebra

Question

In Exercises \(17-26,\) 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 \(X\). $$ 3 X+A=B $$

Step-by-Step Solution

Answer

The solution for matrix X is X = [[-2/3, 2], [-2/3, 3], [-2/3, -4/3]].

Step 1: Isolate the Matrix X

In order to solve for matrix X, it needs to be isolated on one side of the equation. This can be done by subtracting Matrix A from both sides. So, the equation \(3X + A = B\) becomes \(3X = B - A\).

Step 2: Calculate the Result of B - A

Subtract matrix A from matrix B elementwise. The resulting matrix, let's call it C, is found by \(C_{ij} = B_{ij} - A_{ij}\), where \(C_{ij}\), \(B_{ij}\) and \(A_{ij}\) are the elements of matrices C,B and A at row i and column j. This gives us the matrix C = [[-2, 6], [-2, 9], [-2, -4]].

Step 3: Divide by 3

Now from step 1, the equation is \(3X = C\). So, to solve for X, just divide every element in matrix C by 3. This gives us X = [[-2/3, 2], [-2/3, 3], [-2/3, -4/3]].

Key Concepts

Matrix Subtraction

Matrix subtraction is a fundamental operation performed elementwise. To subtract one matrix from another, each element in one matrix is subtracted from its corresponding element in the other matrix. In our exercise, we need to subtract matrix \(A\) from matrix \(B\), leading to the result matrix \(C\).

Let's break down the exercise example:

• The element in the first row, first column of \(B = -5\), and in \(A = -3\), results in \(-5 - (-3) = -5 + 3 = -2\).
• Repeat this for each element: \(B_{ij} - A_{ij} = C_{ij}\).

Essentially, matrix subtraction is as simple as subtracting numbers one by one across the matrices. It's crucial to ensure that both matrices are of the same size, as subtraction cannot occur otherwise. This is why always check the dimensions: they must be identical!

Matrix Division

Matrix division is not as straightforward as subtraction. In linear algebra, traditional division does not apply directly to matrices. However, solving matrix equations like \(3X = C\) involves a form of division. Here, we divide a matrix by a scalar, which is similar to multiplying each element by the reciprocal of the scalar.

In our exercise, we divide matrix \(C\) by 3:

• Take each element \(C_{ij}\) and perform \(C_{ij} / 3\).
• This results in each element being reduced by a factor of the scalar, effectively dividing every element by that scalar value.

This operation helps isolate matrix \(X\), allowing us to solve the matrix equation. Remember, dividing a matrix by a scalar changes the scale of the matrix but not its structure.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It encompasses many fundamental tools that are vital for a broad array of applications.

Matrices are a central component of linear algebra, offering a compact way to handle systems of linear equations, transformations, and more. Some reasons matrices and operations like subtraction and division are essential include:

• Simplifying complex systems: Breaks down systems of equations into simpler, manageable components.
• Applications in physics, engineering, and computer science: Matrices model transformations, data structures, and computations.
• Facilitating calculations: Techniques such as row reduction and finding determinants help solve linear systems efficiently.

Linear algebra serves as a foundation for advanced mathematical topics, making understanding these basic operations crucial for any student venturing into fields where these tools are applicable.