Problem 13 - Differential Equations and Linear Algebra

Question

If \(A=\left[\begin{array}{cc}2 & -5 \\ 6 & -6\end{array}\right],\) calculate \(A^{2}\) and verify that \(A\) satisfies \(A^{2}+4 A+18 I_{2}=0_{2}\)

Step-by-Step Solution

Answer

We calculated \( A^2=\left[\begin{array}{cc}-26 & 22 \\-24 & 24\end{array}\right] \), \(4A=\left[\begin{array}{cc}8 & -20 \\24 & -24\end{array}\right]\), and \(18I_2=\left[\begin{array}{cc}18 & 0 \\0 & 18\end{array}\right]\). Adding these matrices, we obtained \(A^2+4A+18I_2 = \left[\begin{array}{cc}0 & 2 \\0 & 18\end{array}\right] \neq 0_2\), so the given equation A² + 4A + 18I₂ = 0₂ is not satisfied by the matrix A.

Step 1: Calculate A² (A times A)

To calculate A², we need to multiply matrix A by itself. The general formula for multiplying two 2x2 matrices is: \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix} \] Using this formula, we can compute A²: \[ A^2 =\left[\begin{array}{cc} 2 & -5 \\ 6 & -6 \end{array}\right] \left[\begin{array}{cc} 2 & -5 \\ 6 & -6 \end{array}\right] = \left[\begin{array}{cc} (2 \times 2) + (-5 \times 6) & (2 \times -5) + (-5 \times -6) \\ (6 \times 2) + (-6 \times 6) & (6 \times -5) + (-6 \times -6) \end{array}\right] = \left[\begin{array}{cc} -26 & 22 \\ -24 & 24 \end{array}\right] \]

Step 2: Calculate 4A

Now, we need to calculate 4A, which means multiplying each element of matrix A by 4: \( 4A = \left[\begin{array}{cc} 4 \times 2 & 4 \times -5 \\ 4 \times 6 & 4 \times -6 \end{array}\right] = \left[\begin{array}{cc} 8 & -20 \\ 24 & -24 \end{array}\right] \)

Step 3: Calculate 18I₂

Next, we need to calculate 18 times identity matrix I₂: \( 18I₂ = 18 \times \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right] = \left[\begin{array}{cc} 18 & 0 \\ 0 & 18 \end{array}\right] \)

Step 4: Calculate A² + 4A + 18I₂

Now, we have all the terms needed for the equation A² + 4A + 18I₂. Let's add them all together: \( A^2 + 4A + 18I₂ = \left[\begin{array}{cc} -26 & 22 \\ -24 & 24 \end{array}\right] + \left[\begin{array}{cc} 8 & -20 \\ 24 & -24 \end{array}\right] + \left[\begin{array}{cc} 18 & 0 \\ 0 & 18 \end{array}\right] = \left[\begin{array}{cc} -26+8+18 & 22-20+0 \\ -24+24+0 & 24-24+18 \end{array}\right] = \left[\begin{array}{cc} 0 & 2 \\ 0 & 18 \end{array}\right] \)

Step 5: Verify equation A² + 4A + 18I₂ = 0₂

The result obtained in Step 4 is not the zero matrix 0₂: \( 0₂ = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] \) So, the given equation A² + 4A + 18I₂ = 0₂ is not satisfied by the matrix A.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra. Unlike multiplying regular numbers, the multiplication of matrices follows specific rules. To multiply two matrices, say matrix A and matrix B, the number of columns in matrix A must equal the number of rows in matrix B. Here's a quick guideline to remember:

• The element in the i-th row and j-th column of the resulting matrix is the sum of the products of elements from the i-th row of the first matrix and the j-th column of the second matrix.

To illustrate this with two 2x2 matrices:
If matrix A is:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]and matrix B is:\[\begin{bmatrix}e & f \g & h\end{bmatrix}\]then their product AB will be:\[\begin{bmatrix}ae + bg & af + bh \ce + dg & cf + dh\end{bmatrix}\]In our exercise, multiplying the matrix A by itself, or finding \(A^2\), ensures we follow these rules carefully to transform A into a new matrix.

Identity Matrix

The identity matrix is a special type of matrix in algebra. It's like the '1' in matrix mathematics. When you multiply any matrix by an identity matrix, you get the original matrix back. This property is similar to how multiplying any number by 1 results in the original number.For a 2x2 identity matrix, it looks like this:\[I_2 = \begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]If you multiply any 2x2 matrix by \(I_2\), you end up with the unchanged original matrix. This property is very useful in solving matrix equations because multiplying by the identity matrix doesn't change the outcome.

Zero Matrix

The zero matrix is another important concept in matrix algebra. All its entries are zero, and it acts like the '0' of matrix operations. When you add a zero matrix to any matrix, you get the original matrix back, just like adding zero to a number.For a 2x2 zero matrix, it is written as:\[0_2 = \begin{bmatrix}0 & 0 \0 & 0\end{bmatrix}\]In context, if you multiply any matrix by the zero matrix, the result is also a zero matrix, as each element gets multiplied by zero. This property plays a crucial role in verifying solutions to matrix equations, such as demonstrating that combinations effectively equal zero as in our exercise.

Matrix Equation

A matrix equation is a mathematical statement involving matrices. It often equates one or more expressions on its left side with those on its right side. Solving such equations typically involves operations like matrix addition, subtraction, multiplication, and sometimes finding inverses.

• For example, in an equation like \(A^2 + 4A + 18I_2 = 0_2\), you're being asked to combine and manipulate matrices to see if they meet the condition on the right side (zero matrix in this case).

Understanding each component helps. Break down matrices, separately calculate them as shown in steps, and verify their sums' satisfaction of the equation. Keep track of how each modified matrix or scalar multiple contributes to the final overall result.