Problem 30 - Algebra for College Students

Question

For Problems \(27-30\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} -2 & 3 \\ 5 & 4 \end{array}\right] \quad B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right] \quad D=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\\\ &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}\) $$ \text { Compute } A I \text { and } I A \text {. } $$

Step-by-Step Solution

Answer

\( A I = A \) and \( I A = A \).

Step 1: Understanding Matrix Multiplication

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix. The resulting matrix will have the dimensions of the outer dimensions of the two matrices being multiplied.

Step 2: Identify Matrices for Multiplication

Here, we need to find \( A I \) and \( I A \). Both involve the identity matrix \( I \), which has the property that any matrix multiplied by the identity matrix remains unchanged.

Step 3: Compute \( A I \)

To compute \( A I \), multiply each element in the rows of matrix \( A \) by the corresponding elements in the columns of matrix \( I \). Given: \[A = \left[\begin{array}{rr} -2 & 3 \ 5 & 4 \end{array}\right], I = \left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right] \]The resulting matrix is: \[A I = \left[\begin{array}{rr} (-2)(1) + (3)(0) & (-2)(0) + (3)(1) \ (5)(1) + (4)(0) & (5)(0) + (4)(1) \end{array}\right] = \left[\begin{array}{rr} -2 & 3 \ 5 & 4 \end{array}\right] \]

Step 4: Compute \( I A \)

Similarly, compute \( I A \) using the same process. Given: \[I = \left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right], A = \left[\begin{array}{rr} -2 & 3 \ 5 & 4 \end{array}\right] \]The resulting matrix is: \[I A = \left[\begin{array}{rr} (1)(-2) + (0)(5) & (1)(3) + (0)(4) \ (0)(-2) + (1)(5) & (0)(3) + (1)(4) \end{array}\right] = \left[\begin{array}{rr} -2 & 3 \ 5 & 4 \end{array}\right] \]

Step 5: Conclusion

The multiplication of any matrix by the identity matrix results in the original matrix. Thus, both \( A I \) and \( I A \) give the matrix \( A \).

Key Concepts

Identity Matrix

The Identity Matrix is a special type of square matrix that plays a crucial role in matrix algebra. It is analogous to the number 1 in regular multiplication. The identity matrix has a simple structure:

• It is always a square matrix, meaning it has the same number of rows and columns.
• Its main diagonal consists entirely of 1s (ones).
• All other elements are 0s (zeros).

For a 2x2 identity matrix, it looks like this:\[I = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right]\]
When any matrix is multiplied by the identity matrix, it remains unchanged. This property makes the identity matrix a powerful tool in matrix operations, ensuring that the original matrix characteristics are preserved in the product. It is also used to find inverses and solve matrix equations effectively.

Matrix Algebra

Matrix Algebra is a branch of mathematics that deals with operations on matrices. It covers different ways to handle matrices, including addition, subtraction, and multiplication.
In matrix multiplication, the following rules are pivotal:

• Only matrices with compatible dimensions can be multiplied. This means the number of columns in the first matrix must match the number of rows in the second matrix.
• The order of multiplication matters; the product of two matrices isn't necessarily commutative, meaning \(AB eq BA\).
• The resulting matrix will have dimensions equal to the outer dimensions of the matrices being multiplied. For instance, multiplying a 2x3 matrix by a 3x2 matrix results in a 2x2 matrix.

Comprehending these rules helps in effectively performing matrix operations and applying these skills to solve real-world problems or theoretical exercises in fields like engineering and physics.

Properties of Matrices

Matrices have unique properties that influence how they interact with each other. Understanding these properties can simplify complex problems and ensure accurate results when performing calculations. Some of the key properties include:

• Associative Property: For matrix multiplication, the associative property holds, meaning \((AB)C = A(BC)\).
• Distributive Property: Matrices obey the distributive laws such that \(A(B + C) = AB + AC\).
• Identity Property: As already discussed, multiplying any matrix by the identity matrix returns the original matrix, i.e., \(AI = IA = A\).
• Zero Matrix: Similar to "zero" in arithmetic, multiplying any matrix by a zero matrix results in a zero matrix.

Understanding these properties is essential for dealing with matrix equations and applications. They serve as guidelines to ensure the correct manipulation of matrices and allow for systematic solutions to complex mathematical problems.