Problem 11 - Algebra 2

Question

Solve each matrix equation. $$ \left[\begin{array}{rrr}{6} & {10} & {-13} \\ {4} & {-2} & {7} \\ {0} & {9} & {-8}\end{array}\right] X=\left[\begin{array}{l}{84} \\ {18} \\\ {56}\end{array}\right] $$

Step-by-Step Solution

Answer

To solve for \( X \) multiply both sides of the equation by \( A^{-1} \) if it exists, \( X = A^{-1}B \) and then calculate \( X \) by matrix multiplication.

Step 1 - Setting up the equation

Let the matrix \( A = \left[\begin{array}{rrr}{6} & {10} & {-13} \ {4} & {-2} & {7} \ {0} & {9} & {-8}\end{array}\right] \) and the matrix \( B = \left[\begin{array}{l}{84} \ {18} \ {56}\end{array}\right] \). The matrix equation to solve is \( AX = B \) where \( X \) is the matrix we are solving for.

Step 2 - Find the inverse of matrix A

To find \( X \), we need to calculate the inverse of \( A \), denoted as \( A^{-1} \). If \( A^{-1} \) exists, we can multiply both sides of the equation by it: \( A^{-1}AX = A^{-1}B \), which simplifies to \( X = A^{-1}B \) because \( A^{-1}A \) is the identity matrix.

Step 3 - Calculate the inverse of matrix A (if possible)

This step involves computation of the inverse through methods such as Gaussian elimination, finding the adjugate and dividing by the determinant, or utilizing a technology aid. This computation is generally non-trivial and can be extensive.

Step 4 - Multiply the inverse of matrix A by matrix B

Once we find the inverse \( A^{-1} \), we then multiply it with matrix \( B \) to find \( X \). This step involves matrix multiplication where each entry \( ij \) of the resulting matrix \( X \) is the dot product of the i-th row of \( A^{-1} \) and the j-th column of \( B \)

Key Concepts

Inverse of a Matrix

Understanding the concept of the inverse of a matrix is crucial for solving matrix equations. Suppose we have a square matrix, denoted by the letter 'A'. The inverse of this matrix, if it exists, is another matrix, say 'B', such that when 'A' is multiplied by 'B', the result is an identity matrix. An identity matrix is a special kind of matrix with 1s on the diagonal and 0s elsewhere, and it acts somewhat like the number 1 in regular multiplication.

The process to find the inverse of a matrix, denoted as 'A^-1', requires that the matrix be non-singular, which means it must have a non-zero determinant. To compute the inverse, one of the most common methods is to use Gaussian elimination or to find the adjugate of the matrix and divide it by the determinant. The existence of an inverse is fundamental to solve equations of the form 'AX = B', because when we multiply both sides of the equation by 'A^-1', we get 'X = A^-1B', isolating our variable 'X'.

Matrix Multiplication

Matrix multiplication is a key operation in solving matrix equations. While it might be tempting to think of it as similar to multiplying numbers, it's quite different. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second. To get one entry of the product matrix, you take the dot product of one row of the first matrix with one column of the second matrix.

Here's a simpler visualization: you multiply each element of the row by its corresponding element in the column, and then add all those products together. This sum is the value of one entry in the resulting matrix. Now, repeat this process for each entry. Remember, matrix multiplication is not commutative, meaning that 'AB' may not equal 'BA'. This characteristic is crucial to remember when finding solutions to matrix equations like 'X = A^-1B'.

Gaussian Elimination

Gaussian elimination is a systematic method used for solving systems of linear equations, and it's also used in finding the inverse of a matrix and the determinant. It involves performing operations on the rows of a matrix to achieve what we call row echelon form. A matrix is in row echelon form when all non-zero rows are above rows filled with zeroes, and the leading coefficient (also known as the pivot) of a non-zero row is always strictly to the right of the leading coefficient of the row above it.

The operations used in Gaussian elimination are swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row to another. The aim is to systematically eliminate the variables and solve the equation step by step. When applied to finding an inverse, the process is augmented by applying these operations to the identity matrix alongside the original matrix. Successfully following this method results in transforming the original matrix into the identity matrix and the identity matrix into the inverse of the original matrix.

Adjugate of a Matrix

The adjugate, or sometimes called the adjoint, of a matrix is another matrix that plays a crucial role in calculating the inverse. To find the adjugate of a matrix 'A', we need to calculate the cofactors of 'A' and then create a matrix of these cofactors, which is known as the cofactor matrix. Then, the adjugate is simply the transpose of this cofactor matrix. The transpose of a matrix is made by swapping the rows and columns of the original matrix.

Each cofactor is calculated by taking the determinant of a smaller matrix that comes from 'A' by removing the row and column of a particular element. This might sound complicated, but it's like zooming in on the matrix, cutting out one row and column and seeing what's left. Significantly, the adjugate can be used to calculate the inverse of a matrix as it is related to the determinant. If the determinant is not zero, the inverse of 'A' can be found by dividing the adjugate by the determinant. This process is an alternative to using Gaussian elimination, especially when working with larger matrices where manual computation might be cumbersome.