Problem 64 - Precalculus: Mathematics for Calculus

Question

No Zero-Product Property for Matrices We have eused the Zero-Product Property to solve algebraic equations. Matrices do not have this property. Let \(O\) represent the \(2 \times 2\) zero matrix $$O=\left[\begin{array}{ll}0 & 0 \\\0 & 0\end{array}\right]$$ Find \(2 \times 2\) matrices \(A \neq O\) and \(B \neq O\) such that \(A B=O\) ECan you find a matrix \(A \neq O\) such that \(A^{2}=O ?\)

Step-by-Step Solution

Answer

Yes, matrices \( A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \) have \( AB = O \), and \( A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \) has \( A^2 = O \).

Step 1: Understand the Zero Matrix

In this problem, we want to find matrices that multiply to yield the zero matrix \( O \). The zero matrix \( O \) is a matrix with all elements equal to zero. Here, \( O \) is \( \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \).

Step 2: Find Matrices A and B such that AB = O

We need to find matrices \( A \) and \( B \), both \( 2 \times 2 \) matrices, such that their product is \( O \). One approach is to choose matrices with distinct patterns but result in zero elements throughout after multiplication. Consider \( A = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \). Perform the matrix multiplication:\[ AB = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \]Thus, \( AB = O \).

Step 3: Check A ≠ O and B ≠ O

Ensure that neither \( A \) nor \( B \) is the zero matrix. For the chosen matrices:- \( A = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \), which is not the zero matrix because it has a '1'.- \( B = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \), which is not the zero matrix because it has a '1'.Both matrices satisfy \( A eq O \) and \( B eq O \).

Step 4: Find Matrix A Such That A² = O

Now, find a matrix \( A \), not equal to \( O \), such that \( A^2 = O \). Consider \( A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \). Check its square:\[ A^2 = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \]Thus, \( A^2 = O \).

Step 5: Verify A ≠ O for A Such That A² = O

Finally, confirm that the matrix \( A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \) is not the zero matrix. It contains a '1', so it is not the zero matrix.

Key Concepts

matrix multiplication

Matrix multiplication is a fundamental operation in linear algebra, involving two matrices to produce another matrix. Unlike scalar multiplication, matrix multiplication is not commutative, meaning that the order of the matrices matters. If you have two matrices \( A \) and \( B \), it is not always true that \( AB = BA \). To multiply matrices, you take the rows of the first matrix, and multiply them by the columns of the second matrix. This involves calculating the dot product of the rows and columns:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• The resulting matrix has dimensions consisting of the number of rows of the first matrix and the number of columns of the second matrix.
• Each element in the resulting matrix is computed by summing the products of elements across the chosen row (from the first matrix) and the corresponding column (from the second matrix).

The process can result in matrices with all zero elements, known as zero matrices, even when both original matrices are not zero matrices. This is a crucial concept when exploring properties such as the zero-product property in matrices.

zero matrix

A zero matrix, denoted as \( O \), is a matrix where all the elements are zeros. This matrix serves as the additive identity in matrix addition, meaning any matrix \( A \) added to a zero matrix of the same dimensions remains unchanged: \( A + O = A \).The zero matrix is significant in multiplication exercises for matrices, as the zero-product property does not hold in matrix mathematics. For example, two non-zero matrices can have a product that is a zero matrix. Consider matrices \( A \) and \( B \):

• If \( A = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \)
• And \( B = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \)

You find that \( AB = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = O \), even though neither \( A \) nor \( B \) is a zero matrix. This exemplifies that in matrix algebra, zero matrices can appear through products of non-zero matrices, contrasting scalar zero-product principles.

non-zero matrices

Non-zero matrices are matrices where at least one element is not zero. They play a vital role in illustrating why matrices do not possess a zero-product property like regular numbers.To find an example where two non-zero matrices multiply to give a zero matrix, consider:

• Matrix \( A = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \)
• Matrix \( B = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \)

While both matrices have elements that are non-zero, their product is the zero matrix \( O \):\[AB = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\]Another example of a non-zero matrix producing the zero matrix when squared is:

• Matrix \( A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \)

With \( A \cdot A = O \), despite matrix \( A \) itself not being a zero matrix.These examples highlight the unique behavior of matrix mathematics and emphasize the importance of understanding how non-zero matrices can interact to result in a zero matrix.