Construct the matrix $A=[a_{ij}]_{3\times3}$, where $a_{ij}=i-j$. State whether $A$ is symmetric or skew-symmetric.

Construct the matrix A=[aᵢⱼ]3×3, where aᵢⱼ=i-j. State whether A is symmetric or skew-symmetric.

Exercise 7.1 · Q21

A=[a_{ij}]_{3\times3}

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We evaluate $a_{ij}=i-j$ for every pair $(i,j)$ from $1$ to $3$ to build the matrix, then check the symmetric/skew-symmetric condition directly on the formula.

Step 1. Compute every entry using $a_{ij}=i-j$ .

$a_{11}=1-1=0,\ a_{12}=1-2=-1,\ a_{13}=1-3=-2$

$a_{21}=2-1=1,\ a_{22}=2-2=0,\ a_{23}=2-3=-1$

$a_{31}=3-1=2,\ a_{32}=3-2=1,\ a_{33}=3-3=0$

Step 2. Assemble the matrix.

A=\begin{pmatrix}0&-1&-2\\1&0&-1\\2&1&0\end{pmatrix}

Step 3. Test the symmetric condition ( $a_{ij}=a_{ji}$ ). E.g. $a_{12}=-1$ but $a_{21}=1$ — these are NOT equal, so $A$ is not symmetric. …

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Q.Construct the matrix $A=[a_{ij}]_{3\times3}$ , where $a_{ij}=i-j$ . State whether $A$ is symmetric or skew-symmetric.

Concept understanding — Symmetric And Skew Symmetric Matrices

Symmetric and Skew-Symmetric Matrices

Symmetric matrix

Skew-symmetric matrix

Key facts

Q.Construct the matrix A=[aij]3×3A=[a_{ij}]_{3\times3}A=[aij​]3×3​, where aij=i−ja_{ij}=i-jaij​=i−j. State whether AAA is symmetric or skew-symmetric.

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Q.Construct the matrix $A=[a_{ij}]_{3\times3}$ , where $a_{ij}=i-j$ . State whether $A$ is symmetric or skew-symmetric.

Unlock everything free for 14 days