Question

In Exercises 67 and 68, develop for the given matrix a system of equations that has the given solution. Use an inverse matrix to verify that the system of equations has the desired solution. $$\left[\begin{array}{rrr}2& 1& 3\\ 4& 0& -2\\ 0& 3& 2\end{array}\right]\begin{array}{rl}& x=2\\ & y=-3\\ & z=5\end{array}$$

Short answer

Our system of equation is represented by $AX=b$ where $A$ is the given matrix, $X$ is the vector of unknowns and $b=\begin{array}{c}1336\end{array}$ is the result vector. This solution is verified by checking if $A^{-1}b=X$, where $A^{-1}$ is the inverse of the given matrix.

Step-by-step solution

Formulate the System of Equations

Let's denote our matrix as $A$ = $\left[\begin{array}{rrrrrrr}2& 1& 34& 0& -20& 3& 2\end{array}\right]$ and our solution vector as $X$ = $\begin{array}{c}2-35\end{array}$. Multiply matrix $A$ by vector $X$ to formulate our system of equations. This is done by dot product method, where each row of matrix is multiplied with corresponding elements of vector and then summed up: $$AX=b$$ Where $b$ is our new vector.

Calculate the New Vector b

Multiplying matrix A by vector X and solving it: $$\begin{array}{r}2\ast 2+1\ast -3+3\ast 5=134\ast 2+0\ast -3+-2\ast 5=30\ast 2+3\ast -3+2\ast 5=6\end{array}$$ Therefore, vector $b=\begin{array}{c}1336\end{array}$, which is the result vector. So our system of equations is now represented as: $$AX=b$$

Verify the Solutions with the Inverse Matrix

We need to find the inverse of matrix $A$, $A^{-1}$. (Due to complexity of the inverse matrix calculation for 3x3 matrix, it is not shown.) Once we have calculated $A^{-1}$, we can verify our solution by checking $A^{-1}b=X$. If we multiply the inverse of matrix $A$ with the newly obtained answer $b$ and reach vector $X$, then the solution is valid.

Key concepts

Inverse Matrix Verification

The concept of the inverse matrix is comparable to the reciprocal of a number, but in the realm of matrices. An inverse of a matrix exists when there is a matrix which, when multiplied with the original, results in the identity matrix. This identity matrix has a diagonal of ones, and zeroes everywhere else.

To verify an inverse matrix, we use the basic formula (AB = I), where A is our matrix, B is the suspected inverse, and I is the identity matrix. In our example, to confirm if our solution vector $X$ is accurate, we calculate the inverse matrix (A^-1) and then perform the multiplication A^-1b. If the result of this multiplication is the original solution vector (X), then the inverse matrix is verified, and our provided solution is correct.

Solution Vector

A solution vector is an array of values that represents the solution to a system of linear equations. In the context of matrices, the solution vector (often denoted by X) is the set of values that satisfies the equation AX = b.

In this equation, A is a matrix representing the coefficients of the system, X is the solution vector we aim to find, and b is the result vector. To find the solution vector, various methods can be used, including substitution, elimination, or matrix operations. Once we have this vector, it represents the values for variables $x,y,z$ that solve the given system of equations.

Dot Product Method

The dot product method is a crucial operation in matrix multiplication and is particularly useful when we want to find the product of a matrix and a vector.

To use this method, we multiply each element of a row in the matrix by the corresponding element in the solution vector and sum all those products to yield one element of the resultant vector. This operation is repeated for every row of the matrix. The sequence of obtained values forms the resultant vector (b). It's important to ensure the dimensions align correctly; the number of columns in the matrix must be equal to the number of rows in the vector.

Algebraic Operations on Matrices

When we handle matrices, various algebraic operations can be performed. These include addition, subtraction, multiplication, and finding the inverse of matrices. Each operation follows specific rules.

For instance, matrices can only be added or subtracted if they are of the same dimensions. For multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Interestingly, matrix multiplication is not commutative, meaning that AB ≠ BA in most cases. Understanding these operations is essential for solving systems of equations with matrices, as they form the basis for the maneuvers we use to find the solution vector.