Solving for the Missing Entry

Oct 13, 2025 by windailys.com 57 views

Introduction

Matrix equations are a fundamental concept in linear algebra, and they have numerous applications in various fields, including physics, engineering, and computer science. In this article, we will focus on solving matrix equations, specifically finding the value of the missing entry in a given matrix equation.

What is a Matrix Equation?

A matrix equation is an equation that involves matrices, which are rectangular arrays of numbers. Matrix equations are used to represent systems of linear equations, and they can be solved using various methods, including substitution, elimination, and matrix inversion.

The Given Matrix Equation

The given matrix equation is:

\left[\begin{array}{ll}1 \frac{1}{2} & 1 / 4 \\ \frac{1}{2} & 1 / 3\end{array}\right] + \left[\begin{array}{ll}\square & 1 / 4 \\ 1 / 3 & 1\end{array}\right] = \left[\begin{array}{cc}2 & 1/2 \\ 5/6 & 7/6\end{array}\right]

Step 1: Understand the Problem

The problem requires us to find the value of the missing entry in the matrix equation. To do this, we need to understand the concept of matrix addition and how it applies to the given equation.

Matrix Addition

Matrix addition is a fundamental operation in linear algebra that involves adding corresponding elements of two matrices. The resulting matrix has the same dimensions as the original matrices, and each element is the sum of the corresponding elements in the two matrices.

Step 2: Add the Matrices

To solve the problem, we need to add the two matrices on the left-hand side of the equation. This will give us a new matrix that we can compare with the matrix on the right-hand side.

\left[\begin{array}{ll}1 \frac{1}{2} & 1 / 4 \\ \frac{1}{2} & 1 / 3\end{array}\right] + \left[\begin{array}{ll}\square & 1 / 4 \\ 1 / 3 & 1\end{array}\right] = \left[\begin{array}{ll}1 \frac{1}{2} + \square & 1 / 4 + 1 / 4 \\ \frac{1}{2} + 1 / 3 & 1 / 3 + 1\end{array}\right]

Step 3: Simplify the Equation

Now that we have added the matrices, we can simplify the equation by combining like terms.

\left[\begin{array}{ll}1 \frac{1}{2} + \square & 1 / 2 \\ \frac{5}{6} & 4/3\end{array}\right] = \left[\begin{array}{cc}2 & 1/2 \\ 5/6 & 7/6\end{array}\right]

Step 4: Find the Missing Entry

The final step is to find the value of the missing entry in the matrix equation. To do this, we need to equate the corresponding elements of the two matrices.

1 \frac{1}{2} + \square = 2

\frac{5}{6} = \frac{5}{6}

\frac{1}{2} = \frac{1}{2}

\frac{4}{3} = \frac{7}{6}

Solving for the Missing Entry

Now that we have equated the corresponding elements of the two matrices, we can solve for the missing entry.

1 \frac{1}{2} + \square = 2

\square = 2 - 1 \frac{1}{2}

\square = 2 - \frac{3}{2}

\square = \frac{4}{2} - \frac{3}{2}

\square = \frac{1}{2}

Conclusion

In this article, we have solved a matrix equation by finding the value of the missing entry. We started by understanding the concept of matrix addition and how it applies to the given equation. We then added the two matrices on the left-hand side of the equation and simplified the resulting matrix. Finally, we equated the corresponding elements of the two matrices and solved for the missing entry.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$ .

References

[1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
[2] Matrix Algebra, 2nd Edition, by James E. Gentle

Additional Resources

[1] Khan Academy: Linear Algebra
[2] MIT OpenCourseWare: Linear Algebra
[3] Wolfram MathWorld: Matrix Addition
Solving Matrix Equations: A Q&A Guide =====================================

Introduction

In our previous article, we solved a matrix equation by finding the value of the missing entry. In this article, we will provide a Q&A guide to help you understand the concept of matrix equations and how to solve them.

Q: What is a matrix equation?

A: A matrix equation is an equation that involves matrices, which are rectangular arrays of numbers. Matrix equations are used to represent systems of linear equations, and they can be solved using various methods, including substitution, elimination, and matrix inversion.

Q: What is the difference between matrix addition and matrix multiplication?

A: Matrix addition involves adding corresponding elements of two matrices, while matrix multiplication involves multiplying corresponding elements of two matrices and summing the results.

Q: How do I add matrices?

A: To add matrices, you need to add corresponding elements of the two matrices. For example, if you have two matrices:

\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] + \left[\begin{array}{ll}e & f \\ g & h\end{array}\right]

You would add the corresponding elements:

\left[\begin{array}{ll}a + e & b + f \\ c + g & d + h\end{array}\right]

Q: How do I multiply matrices?

A: To multiply matrices, you need to multiply corresponding elements of the two matrices and sum the results. For example, if you have two matrices:

\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \times \left[\begin{array}{ll}e & f \\ g & h\end{array}\right]

You would multiply the corresponding elements and sum the results:

\left[\begin{array}{ll}ae + bg & af + bh \\ ce + dg & cf + dh\end{array}\right]

Q: What is the inverse of a matrix?

A: The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. The inverse of a matrix is denoted by the symbol $A^{-1}$ .

Q: How do I find the inverse of a matrix?

A: To find the inverse of a matrix, you need to use a method such as Gaussian elimination or LU decomposition. These methods involve transforming the matrix into a form where the inverse can be easily calculated.

Q: What is the identity matrix?

A: The identity matrix is a matrix that has 1s on the main diagonal and 0s elsewhere. The identity matrix is denoted by the symbol $I$ .

Q: How do I use the identity matrix to solve a matrix equation?

A: To use the identity matrix to solve a matrix equation, you need to multiply both sides of the equation by the inverse of the matrix. This will result in the solution to the equation.

Q: What are some common applications of matrix equations?

A: Matrix equations have numerous applications in various fields, including physics, engineering, and computer science. Some common applications include:

Solving systems of linear equations
Finding the inverse of a matrix
Calculating the determinant of a matrix
Solving matrix equations with multiple variables

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of matrix equations and how to solve them. We have covered topics such as matrix addition, matrix multiplication, the inverse of a matrix, and the identity matrix. We have also discussed some common applications of matrix equations.

Final Answer

The final answer is that matrix equations are a powerful tool for solving systems of linear equations and have numerous applications in various fields.

References

[1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
[2] Matrix Algebra, 2nd Edition, by James E. Gentle
[3] Khan Academy: Linear Algebra
[4] MIT OpenCourseWare: Linear Algebra
[5] Wolfram MathWorld: Matrix Addition