If \( A \) is a square matrix such that \( A^2 = 2I \), then the value of \( (A - I)^2 + (A + I)^2 - 7A \) is:
\( -7A + 6I \)
Understanding the Matrix Expression Problem
The problem asks us to simplify a given matrix expression: \( (A - I)^2 + (A + I)^2 - 7A \). We are given a crucial condition that the square matrix \( A \) satisfies \( A^2 = 2I \), where \( I \) represents the identity matrix. We need to use this condition to simplify the expression and find which of the given options is equivalent to the simplified form.
Key Matrix Properties Used
To solve this problem, we will use the following properties of matrix operations:
- Matrix multiplication is generally not commutative (\( AB \neq BA \)), but a matrix commutes with the identity matrix: \( AI = IA = A \).
- Matrix addition and subtraction are associative and commutative.
- Expanding squared terms: \( (X + Y)^2 = X^2 + XY + YX + Y^2 \) and \( (X - Y)^2 = X^2 - XY - YX + Y^2 \).
- For matrices \( A \) and \( I \), we can expand \( (A - I)^2 \) and \( (A + I)^2 \) similarly to algebraic expressions, keeping in mind the order of multiplication.
Step-by-Step Solution for Matrix Expression Simplification
Step 1: Expand \( (A - I)^2 \)
We expand the term \( (A - I)^2 \):
\( (A - I)^2 = (A - I)(A - I) \)
Using the distributive property:
\( (A - I)^2 = A \cdot A - A \cdot I - I \cdot A + I \cdot I \)
Since \( AI = A \), \( IA = A \), and \( I \cdot I = I^2 = I \):
\( (A - I)^2 = A^2 - A - A + I \)
Combining like terms:
\( (A - I)^2 = A^2 - 2A + I \)
Step 2: Expand \( (A + I)^2 \)
Next, we expand the term \( (A + I)^2 \):
\( (A + I)^2 = (A + I)(A + I) \)
Using the distributive property:
\( (A + I)^2 = A \cdot A + A \cdot I + I \cdot A + I \cdot I \)
Since \( AI = A \), \( IA = A \), and \( I \cdot I = I^2 = I \):
\( (A + I)^2 = A^2 + A + A + I \)
Combining like terms:
\( (A + I)^2 = A^2 + 2A + I \)
Step 3: Add the Expanded Expressions
Now, we add the expanded forms of \( (A - I)^2 \) and \( (A + I)^2 \):
\( (A - I)^2 + (A + I)^2 = (A^2 - 2A + I) + (A^2 + 2A + I) \)
\( = A^2 - 2A + I + A^2 + 2A + I \)
Grouping like terms:
\( = (A^2 + A^2) + (-2A + 2A) + (I + I) \)
\( = 2A^2 + 0A + 2I \)
\( = 2A^2 + 2I \)
Step 4: Use the Given Condition \( A^2 = 2I \)
We are given that \( A^2 = 2I \). We substitute this into the expression \( 2A^2 + 2I \):
\( 2A^2 + 2I = 2(2I) + 2I \)
\( = 4I + 2I \)
\( = 6I \)
So, \( (A - I)^2 + (A + I)^2 = 6I \).
Step 5: Include the Remaining Term \( -7A \)
The full expression is \( (A - I)^2 + (A + I)^2 - 7A \). We substitute the result from Step 4:
\( (A - I)^2 + (A + I)^2 - 7A = (6I) - 7A \)
\( = 6I - 7A \)
This can also be written as \( -7A + 6I \).
Step 6: Compare with Options
We compare our simplified expression \( -7A + 6I \) with the given options:
- Option 1: \( 2I \)
- Option 2: \( -7A + 6I \)
- Option 3: \( -8I - 7A \)
- Option 4: \( 8I + 7A \)
Our result matches Option 2.
Conclusion
By expanding the squared terms, combining like terms, and substituting the given condition \( A^2 = 2I \), we simplified the expression \( (A - I)^2 + (A + I)^2 - 7A \) to \( -7A + 6I \).
| Step | Operation | Expression |
|---|---|---|
| 1 | Expand \( (A - I)^2 \) | \( A^2 - 2A + I \) |
| 2 | Expand \( (A + I)^2 \) | \( A^2 + 2A + I \) |
| 3 | Add expansions | \( (A^2 - 2A + I) + (A^2 + 2A + I) = 2A^2 + 2I \) |
| 4 | Substitute \( A^2 = 2I \) | \( 2(2I) + 2I = 4I + 2I = 6I \) |
| 5 | Subtract \( 7A \) | \( 6I - 7A \) |
Revision Table: Key Concepts in Matrix Algebra
| Concept | Description | Property Used Here |
|---|---|---|
| Square Matrix | A matrix with the same number of rows and columns. | \( A \) is a square matrix. |
| Identity Matrix (\( I \)) | A square matrix with 1s on the main diagonal and 0s elsewhere. Acts like the number 1 in multiplication (\( AI = IA = A \)). | Commutes with \( A \), \( I^2 = I \). |
| Matrix Addition/Subtraction | Performed element-wise for matrices of the same dimensions. Commutative and associative. | Combining terms like \( A^2+A^2 \) or \( -2A+2A \). |
| Matrix Multiplication | A more complex operation. Generally not commutative (\( AB \neq BA \)). Distributive over addition/subtraction. | Expanding \( (A-I)^2 \) and \( (A+I)^2 \). |
Additional Information: Working with Identity Matrices
The identity matrix \( I \) plays a role similar to the number 1 in matrix algebra. For any matrix \( A \) for which the products are defined, \( AI = A \) and \( IA = A \). This property was essential in simplifying the expansions of \( (A-I)^2 \) and \( (A+I)^2 \).
When expanding expressions like \( (A+I)^2 \), it's important to remember the order of matrix multiplication. \( (A+I)^2 = (A+I)(A+I) = AA + AI + IA + II \). If the matrices commute (like \( A \) and \( I \) do, or if \( A \) was a scalar multiple of \( I \)), then \( AI+IA = A+A = 2A \). However, if \( A \) and \( I \) did not commute (which is not possible for the identity matrix, but consider general matrices \( A \) and \( B \)), then \( (A+B)^2 = A^2 + AB + BA + B^2 \), and \( AB+BA \) cannot be combined into \( 2AB \) unless \( AB=BA \).
In this problem, the identity matrix simplifies the expansions greatly:
- \( (A - I)^2 = A^2 - AI - IA + I^2 = A^2 - A - A + I = A^2 - 2A + I \)
- \( (A + I)^2 = A^2 + AI + IA + I^2 = A^2 + A + A + I = A^2 + 2A + I \)
The given condition \( A^2 = 2I \) was then used as a direct substitution to further simplify the expression, leading to the final result \( -7A + 6I \).
Similar Questions
If \( A = \begin{bmatrix} x & -y \\ y & x \end{bmatrix} \) and \( A + A' = 2I \), then \( x \) is equal to:
If \( A \) is a matrix and \( |A| \neq 0 \), then the solution of the equation \( XA = B \) is:
If \( A = \begin{bmatrix} 0 & -3 & 2 \\ a & b & 8 \\ -2 & c & 0 \end{bmatrix} \) is a skew-symmetric matrix, then:
Important Questions from Matrices
Three defective bulbs are mixed with 8 good ones. If three bulbs are drawn one by one with replacement, the probabilities of getting exactly 1 defective, more than 2 defective, no defective and more than 1 defective respectively are :
If A, B, and C are three singular matrices given by:
\[ A = \begin{bmatrix} 1 & 4 \\ 3 & 2a \end{bmatrix} \]
\[ B = \begin{bmatrix} 3b & 5 \\ a & 2 \end{bmatrix} \]
and
\[ C = \begin{bmatrix} a + b + c & c + 1 \\ a + c & c \end{bmatrix} \]
then the value of abc is:
If \( P = \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix} \) and \( Q = \begin{bmatrix} 2 & -4 & 1 \end{bmatrix} \) are two matrices, then \( (PQ)' \) will be:
The matrix
\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
is a:
- \( \text{Scalar matrix} \)
- \( \text{Diagonal matrix} \)
- \( \text{Skew-symmetric matrix} \)
- \( \text{Symmetric matrix} \)
Choose the correct answer from the options given below:
If A is a square matrix and I is an identity matrix such that \(A^2 = A\), then \(A(I - 2A)^3 + 2A^3\) is equal to :
