Problem 45 - Algebra 2

Question

Give an example of two matrices whose sum is a zero matrix.

Step-by-Step Solution

Answer

Matrix 1: \(\begin{bmatrix} 2 & -3 \\ 4 & 1 \end{bmatrix}\) and Matrix 2: \(\begin{bmatrix} -2 & 3 \\ -4 & -1 \end{bmatrix}\).

Step 1: Understand Zero Matrix

A zero matrix is a matrix in which all the elements are zero. It serves as the additive identity for matrices, meaning any matrix added to a zero matrix will result in the original matrix.

Step 2: Define Two Matrices

To find two matrices whose sum is a zero matrix, choose two matrices of the same size such that each corresponding element adds up to zero.

Step 3: Select Specific Matrices

Consider two 2x2 matrices:Matrix 1: \(\begin{bmatrix}2 & -3 \4 & 1\end{bmatrix}\)Matrix 2: \(\begin{bmatrix}-2 & 3 \-4 & -1\end{bmatrix}\)

Step 4: Add the Matrices

Add the two matrices element-wise:\[\begin{bmatrix}2 & -3 \4 & 1\end{bmatrix}+\begin{bmatrix}-2 & 3 \-4 & -1\end{bmatrix}=\begin{bmatrix}2 + (-2) & -3 + 3 \4 + (-4) & 1 + (-1)\end{bmatrix}=\begin{bmatrix}0 & 0 \0 & 0\end{bmatrix}\]

Step 5: Confirm the Result

The resulting matrix is a zero matrix:\(\begin{bmatrix}0 & 0 \0 & 0\end{bmatrix}\)

Step 6: Conclusion

The sum of the selected matrices results in a zero matrix, confirming that these two matrices meet the exercise's criteria.

Key Concepts

Zero Matrix

A zero matrix is one of the simplest types of matrices you'll encounter. Every entry or element of this matrix is the number zero. No matter the size of the matrix, each position contains a zero.
This matrix is crucial because it acts like the number zero does in regular arithmetic.

• For example, adding zero to any number results in the original number.
• Similarly, if you add a zero matrix to any given matrix, the result is the original matrix unchanged.

Therefore, when you see a zero matrix, you can think of it as adding nothing to the equation. It's the additive identity in the vast world of matrices, ensuring they behave nicely under addition.

Additive Identity

The concept of an additive identity is not exclusive to matrices. It's a broad mathematical idea that helps simplify addition operations across various structures. The zero matrix serves as the additive identity for matrices.
In other words, when you add a zero matrix to any matrix \(A\), the resulting sum is \(A\) itself. This property is very much like the role of zero in basic arithmetic where adding zero to any number leaves it unchanged.

• If \(A\) is any matrix, then \(A + 0 = A\).
• This zero matrix identity showcases how the matrix remains intact despite the inclusion of the zero matrix in the operation.

The consistency of this identity ensures stability when working with different-sized matrices and their interactions with the zero matrix.

Element-wise Addition

Matrix addition is a straightforward procedure because of its element-wise nature. When adding matrices, unlike adding numbers, you do it piece by piece. Each element of one matrix is added to the corresponding element in the other matrix.

• For two matrices, \(A = [a_{ij}]\) and \(B = [b_{ij}]\), their sum \(C = A + B\) is another matrix where each element \(c_{ij} = a_{ij} + b_{ij}\).
• This means every position in the resulting matrix is the sum of the elements from the same position in the two original matrices.

Element-wise addition is essential because it ensures that both matrices must be of the same size. This requirement keeps calculations clear and manageable, similar to how you would line up numbers when adding in arithmetic.

Matrix Properties

Understanding matrix properties is key to mastering matrix operations like addition. Matrices have specific rules and characteristics that you must follow.

• One important property is the necessity of matrices being of the same size for addition.
• Another is the commutative property: \(A + B = B + A\) (order doesn't change the sum).
• There is also the associative property which allows grouping: \((A + B) + C = A + (B + C)\).

These properties make it easier to manipulate and simplify matrix expressions just like you would with regular numbers. Plus, when you know the zero matrix is the identity for addition, it simplifies your work, providing a sense of structure and predictability in more complex operations.