Let $A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix}$ and $B$ be a matrix such that $B(I – A) = I + A$. Then the sum of the diagonal elements of $B^T B$ is equal to ________ .
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Solution Steps
1
Identify Property of Matrix A
Observe matrix $A$: $A^T = \begin{bmatrix} 0 & -2 & 3 \\ 2 & 0 & -1 \\ -3 & 1 & 0 \end{bmatrix} = -A$.
Thus, $A$ is a skew-symmetric matrix.
2
Express Matrix B
From the given equation: $B(I – A) = I + A$.
Since $(I – A)$ is non-singular for any skew-symmetric matrix $A$, we can write:
$B = (I + A)(I – A)^{-1}$.
3
Find Transpose of B
$B^T = ((I + A)(I – A)^{-1})^T = ((I – A)^{-1})^T (I + A)^T$.
Using $(M^{-1})^T = (M^T)^{-1}$ and $A^T = -A$:
$B^T = (I – A^T)^{-1} (I + A^T) = (I – (-A))^{-1} (I – A) = (I + A)^{-1} (I – A)$.
4
Calculate $B^T B$
$B^T B = [(I + A)^{-1} (I – A)] \times [(I + A) (I – A)^{-1}]$.
Since $(I – A)$ and $(I + A)$ commute (as both are polynomials in $A$):
$B^T B = (I + A)^{-1} (I + A) (I – A) (I – A)^{-1} = I \cdot I = I$.
This shows $B$ is an orthogonal matrix.
5
Sum of Diagonal Elements
The sum of diagonal elements of a matrix is its Trace, $Tr(M)$.
$Tr(B^T B) = Tr(I)$. Since $A$ is $3 \times 3$, $I$ is the $3 \times 3$ identity matrix.
$Tr(I) = 1 + 1 + 1 = 3$.
Final Answer: 3
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Theory
1. Skew-Symmetric Matrices
A square matrix $A$ is skew-symmetric if $A^T = -A$. Key properties include: the diagonal elements are always zero, and the determinant of an odd-order skew-symmetric matrix is always zero. However, $(I – A)$ where $I$ is the identity matrix and $A$ is skew-symmetric is always invertible because the eigenvalues of a skew-symmetric matrix are purely imaginary, so $1$ can never be an eigenvalue.
2. Cayley Transform
The mapping $B = (I + A)(I – A)^{-1}$ is known as the Cayley Transform. It is a fundamental result in linear algebra that if $A$ is skew-symmetric, then its Cayley transform $B$ is an orthogonal matrix ($B^T B = I$). This provides a way to generate orthogonal matrices from skew-symmetric ones without involving trigonometric functions.
3. Trace of a Matrix
The trace of a square matrix, denoted by $Tr(A)$, is the sum of the elements on the main diagonal. It is a linear operator ($Tr(A+B) = Tr(A) + Tr(B)$) and possesses the cyclic property ($Tr(AB) = Tr(BA)$). For an identity matrix $I_n$ of order $n$, the trace is simply $n$. In this problem, because $B$ is orthogonal, $B^T B$ simplifies to the identity matrix of order 3.
4. Commuting Matrices
Two matrices $M$ and $N$ commute if $MN = NM$. Any two polynomials in the same matrix $A$ (such as $I + A$ and $I – A$) always commute with each other. This property is crucial for simplifying expressions like $(I + A)^{-1}(I – A)(I + A)$, allowing the order of multiplication to be rearranged to cancel terms out to the identity matrix.
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FAQs
1
Why is A called skew-symmetric?
Because $a_{ij} = -a_{ji}$ for all $i, j$. For example, $a_{12} = 2$ and $a_{21} = -2$.
2
What if A was a 2×2 matrix?
The logic would remain the same, but the trace of the identity matrix $I_2$ would be 2.
3
Is $(I – A)^{-1}$ always defined?
Yes, for a real skew-symmetric matrix $A$, $(I-A)$ is always invertible.
4
How do we know B is 3×3?
Since $A$ is 3×3 and $I$ must be the same size to be added/subtracted, $B$ must also be 3×3 for the equation to hold.
5
Does $Tr(B^T B) = Tr(B B^T)$?
Yes, the trace is invariant under cyclic permutations, so $Tr(B^T B) = Tr(B B^T)$.
6
What is an orthogonal matrix?
A matrix $B$ such that $B^T B = B B^T = I$. Its rows and columns are orthonormal vectors.
7
Can I solve this by finding B explicitly?
You can, but it is extremely time-consuming. Using properties of skew-symmetric matrices is the intended “JEE shortcut.”
8
What is $Tr(B^T B)$ if B is skew-symmetric?
If $B$ is skew-symmetric, $B^T = -B$, so $B^T B = -B^2$. The trace would then depend on the specific values.
9
Does the diagonal of A always have to be zero?
Yes, for $A^T = -A$, $a_{ii} = -a_{ii} \implies 2a_{ii} = 0 \implies a_{ii} = 0$.
10
Is this question from a specific exam?
This is from the JEE Main 2026 Online session held on 23rd January, Evening Shift.
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