If A = bmatrix 1 & 0 & 2 \\ 2 & 1 & 3 \\ 4 & 5 & 6 bmatrix a) Write the order of A^T . [1] b) What types does matrix form when (A + A^T) ? [1] c) What is the determinate of matrix A, if A is…

a) Order of $A^T$

The order of a matrix is given by the number of rows × number of columns. For matrix $A$, the order is $3\times 3$. The transpose of $A$, denoted as $A^T$, is obtained by interchanging rows and columns. Thus, the order of $A^T$ remains $3\times 3$.

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b) Type of matrix formed by $(A+A^T)$

First, compute $A^T$: $$A^T=\left[\begin{array}{ccc}1& -2& 4\\ 0& 1& 5\\ 2& 3& 6\end{array}\right]$$

Now, compute $A+A^T$: $$A+A^T=\left[\begin{array}{ccc}1+1& 0-2& 2+4\\ -2+0& 1+1& 3+5\\ 4+2& 5+3& 6+6\end{array}\right]=\left[\begin{array}{ccc}2& -2& 6\\ -2& 2& 8\\ 6& 8& 12\end{array}\right]$$

Observe that $(A+A^T{)}^T=A^T+A=A+A^T$. Thus, $A+A^T$ is a symmetric matrix.

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c) Determinant of matrix $A$ if $A$ is singular

A matrix is singular if its determinant is zero. Thus, if $A$ is singular, then: $$|A|=0$$

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d) Formula for finding $A^{-1}$

The inverse of a $3\times 3$ matrix $A$ is given by: $$A^{-1}=\frac{1}{|A|}\cdot \text{adj}(A)$$ where:

• $|A|$ is the determinant of $A$,
• $\text{adj}(A)$ is the adjugate (transpose of the cofactor matrix) of $A$.

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e) Condition under which $A^{-1}$ does not exist

The inverse of a matrix $A$ does not exist if:

• $A$ is singular (i.e., $|A|=0$), or
• $A$ is non-square (i.e., number of rows ≠ number of columns).

For the given matrix $A$, since it is square, the condition reduces to: $$|A|=0$$