Q.If $A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}$, then verify that

Concept and Intuition

Matrix multiplication $PQ$ is defined only when the number of columns of $P$ equals the number of rows of $Q$. Here $A$ is $2\times3$, so $A^T$ is $3\times2$; multiplying $A^T$ ($3\times2$) by $B$ ($2\times3$) gives a valid $3\times3$ product (inner dimensions 2 match), computed entry-by-entry as row·column dot products.

Step-by-Step Solution

1. $A=\begin{bmatrix}1&-1&2\\0&3&4\end{bmatrix}$, so transposing rows/columns: $A^T=\begin{bmatrix}1&0\\-1&3\\2&4\end{bmatrix}$ (3 rows, 2 columns).
2. $B=\begin{bmatrix}4&0&-3\\-1&-2&-3\end{bmatrix}$ (2 rows, 3 columns). Since $A^T$ has 2 columns and $B$ has 2 rows, $A^TB$ is defined and will be $3\times3$.
3. Row 1 of $A^T$ is $(1,0)$. Dot with each column of $B$: col1 $(4,-1)$: $1(4)+0(-1)=4$; col2 $(0,-2)$: $1(0)+0(-2)=0$; col3 $(-3,-3)$: $1(-3)+0(-3)=-3$. So row 1 of the product is $(4,0,-3)$.
4. Row 2 of $A^T$ is $(-1,3)$. Dot: col1: $-1(4)+3(-1)=-4-3=-7$; col2: $-1(0)+3(-2)=-6$; col3: $-1(-3)+3(-3)=3-9=-6$. Row 2: $(-7,-6,-6)$.
5. Row 3 of $A^T$ is $(2,4)$. Dot: col1: $2(4)+4(-1)=8-4=4$; col2: $2(0)+4(-2)=-8$; col3: $2(-3)+4(-3)=-6-12=-18$. Row 3: $(4,-8,-18)$.
6. Assembling: $A^TB = \begin{bmatrix}4&0&-3\\-7&-6&-6\\4&-8&-18\end{bmatrix}$, which matches option (A) exactly.

Common Mistakes

• Trying to compute $A^TB$ using the original dimensions of $A$ (thinking it's undefined) rather than transposing first to get compatible dimensions.
• Row/column mix-ups when reading off entries of $A^T$ from $A$.

ANSWER: A

Concept and Intuition

For any matrix $A$, the entry $(AA^T)_{ij}$ equals (row $i$ of $A$) $\cdot$ (row $j$ of $A$), because transposing turns the columns of $A^T$ back into the rows of $A$. This is much faster than multiplying full matrices out mechanically, and it also guarantees $AA^T$ is symmetric — a useful self-check.

Step-by-Step Solution

1. Rows of $A$: $R_1=(1,-1,2)$, $R_2=(-2,3,-3)$, $R_3=(4,-4,5)$.
2. Compute all pairwise dot products: $R_1\cdot R_1 = 1+1+4=6$ $R_1\cdot R_2 = -2-3-6=-11$ $R_1\cdot R_3 = 4+4+10=18$ $R_2\cdot R_2 = 4+9+9=22$ $R_2\cdot R_3 = -8-12-15=-35$ $R_3\cdot R_3 = 16+16+25=57$ So $AA^T=\begin{pmatrix}6&-11&18\\-11&22&-35\\18&-35&57\end{pmatrix}$ (symmetric, as expected).
3. Compute $A^T=\begin{pmatrix}1&-2&4\\-1&3&-4\\2&-3&5\end{pmatrix}$, then $A+A^T$ entrywise: $(1,1):1+1=2$, $(1,2):-1-2=-3$, $(1,3):2+4=6$ $(2,2):3+3=6$, $(2,3):-3-4=-7$, $(3,3):5+5=10$ So $A+A^T=\begin{pmatrix}2&-3&6\\-3&6&-7\\6&-7&10\end{pmatrix}$.
4. Subtract entrywise: $AA^T-(A+A^T)$: $(1,1):6-2=4$, $(1,2):-11-(-3)=-8$, $(1,3):18-6=12$ $(2,2):22-6=16$, $(2,3):-35-(-7)=-28$ $(3,3):57-10=47$ Giving $\begin{pmatrix}4&-8&12\\-8&16&-28\\12&-28&47\end{pmatrix}$, which is symmetric — consistent with $AA^T$, $A$, $A^T$ all combining symmetrically here.

Common Mistakes

• Sign slips when subtracting negative entries (e.g. $-11-(-3)$ is $-8$, not $-14$).
• Computing $AA^T$ by full matrix multiplication instead of the faster row-dot-product rule, increasing arithmetic-error risk.
• Mixing up $AA^T$ with $A^TA$ — they are generally different matrices.

ANSWER: B

Concept and Intuition

$(kA)^T = kA^T$ for a scalar $k$, and matrix squaring/transposing here is just direct $2\times2$ arithmetic — no shortcut is needed beyond careful multiplication.

Step-by-Step Solution

1. $A=\begin{bmatrix}2&-3\\-4&1\end{bmatrix} \Rightarrow A^T=\begin{bmatrix}2&-4\\-3&1\end{bmatrix}$.
2. Compute $(A^T)^2 = A^T\cdot A^T$:
   • Row1: $[2(2)+(-4)(-3),\ 2(-4)+(-4)(1)] = [4+12,\ -8-4] = [16,-12]$
   • Row2: $[-3(2)+1(-3),\ -3(-4)+1(1)] = [-6-3,\ 12+1] = [-9,13]$ So $(A^T)^2=\begin{bmatrix}16&-12\\-9&13\end{bmatrix}$.
3. Compute $(12A)^T = 12A^T = \begin{bmatrix}24&-48\\-36&12\end{bmatrix}$.
4. Add: $\begin{bmatrix}16+24&-12-48\\-9-36&13+12\end{bmatrix} = \begin{bmatrix}40&-60\\-45&25\end{bmatrix}$.

Common Mistakes

• Squaring $A$ instead of $A^T$ (though they give related but not identical intermediate numbers here due to non-symmetry) — always transpose first as instructed.
• Sign errors while multiplying negative entries during matrix multiplication.

ANSWER: D

Concept and Intuition

Two determinant facts make this immediate: (1) transposing a matrix never changes its determinant, $\det(M^T)=\det(M)$; (2) the determinant of a product of square matrices equals the product of their determinants, $\det(PQ)=\det(P)\det(Q)$. Combining these collapses the given expression to a simple product of the original (non-transposed) determinants.

Step-by-Step Solution

1. $\det(B^T A^T) = \det(B^T)\cdot\det(A^T)$ (product rule).
2. $\det(B^T)=\det(B)$ and $\det(A^T)=\det(A)$ (transpose rule).
3. So $\det(B^TA^T) = \det(B)\cdot\det(A)$.
4. Given $\det(B^TA^T)=-15$ and $\det(A)=5$: $\det(B)\times5=-15 \Rightarrow \det(B)=-3$.

Common Mistakes

• Forgetting the transpose rule and trying to compute $B^TA^T$ as an actual matrix product with unknown entries — the determinant rules make this unnecessary.

ANSWER: B

Concept and Intuition

A matrix satisfying $AA^T = A^TA$ is called normal; this symmetry is exactly what's needed

to let $A^TA$ and $AA^T$ be swapped freely inside an algebraic manipulation. Since $B$ is built

from $A^{-1}$ and $A^T$, testing $BB^T$ naturally produces a chain that can be simplified using

this swap, collapsing to the identity.

Step-by-Step Solution

1. $B = A^{-1}A^T$.
2. $B^T = (A^{-1}A^T)^T = (A^T)^T (A^{-1})^T = A\,(A^{-1})^T$.
3. Since $(A^{-1})^T = (A^T)^{-1}$ (transpose and inverse commute), $B^T = A(A^T)^{-1}$.
4. $BB^T = \left(A^{-1}A^T\right)\left(A(A^T)^{-1}\right) = A^{-1}\big(A^TA\big)(A^T)^{-1}$.
5. Substitute the given condition $A^TA = AA^T$: $BB^T = A^{-1}\big(AA^T\big)(A^T)^{-1} = (A^{-1}A)\big(A^T(A^T)^{-1}\big) = I \cdot I = I$.
6. So $BB^T = I$, i.e. option (B).

Common Mistakes

• Trying to test $AB^T=I$ or $A^TB^T=I$ instead — those don't simplify cleanly and aren't generally true; the normality condition is specifically what makes $BB^T=I$ collapse neatly.

ANSWER: B

Concept and Intuition

If $AA^T = I$, then $A$ is an orthogonal matrix: its rows form an orthonormal set (each row has unit length, and any two distinct rows have zero dot product). Scaling every entry by 3 (as in $3A$) scales row lengths by 3 (so squared length becomes 9) but preserves orthogonality (dot products of different rows of $3A$ are still zero, since they're $9\times$ the original zero dot products).

Step-by-Step Solution

1. Rows of $3A$: $R_1=(1,2,2)$, $R_2=(2,1,-2)$, $R_3=(a,2,b)$.
2. Check $R_1\cdot R_1 = 1+4+4=9$ ✓ and $R_2\cdot R_2=4+1+4=9$ ✓ (consistent with each row having squared length 9).
3. $R_3\cdot R_3 = a^2+4+b^2 = 9 \Rightarrow a^2+b^2=5$.
4. Orthogonality $R_1\cdot R_3=0$: $1\cdot a + 2\cdot 2 + 2\cdot b = a+4+2b=0 \Rightarrow a+2b=-4$.
5. Orthogonality $R_2\cdot R_3=0$: $2\cdot a+1\cdot 2+(-2)\cdot b = 2a+2-2b=0 \Rightarrow a-b=-1 \Rightarrow a=b-1$.
6. Substitute into $a+2b=-4$: $(b-1)+2b=-4 \Rightarrow 3b=-3 \Rightarrow b=-1$, then $a=b-1=-2$.
7. Verify: $a^2+b^2 = 4+1=5$ ✓ (matches step 3).
8. Compute $\dfrac{a}{b}+\dfrac{b}{a} = \dfrac{a^2+b^2}{ab} = \dfrac{5}{(-2)(-1)} = \dfrac{5}{2}$.

Common Mistakes

• Forgetting that $AA^T=I$ constrains the rows (not columns) of $A$ to be orthonormal — using columns instead of rows here would give the wrong equations (though by symmetry of this particular matrix it happens not to matter much, it's important to know which one the condition actually refers to).
• Sign errors when setting up the two linear dot-product equations.

ANSWER: D

Concept and Intuition

When a matrix-relation problem gives several plausible-looking "$S^2$ equals ___" options, it's worth checking whether one of them is actually a general algebraic identity true for any matrix, independent of the specific conditions given — sometimes exam options include exactly this kind of "always true" statement as the intended answer, especially when the other options would require additional unstated assumptions (like orthogonality) to hold. Here, $(S^T)^2 = (S^2)^T$ follows purely from the transpose-reversal rule $(XY)^T = Y^TX^T$, with no special conditions needed at all.

Step-by-Step Solution

1. Treat $A,B,C,D$ as $1\times1$ matrices (scalars) to build a concrete, valid test case (transpose of a scalar is itself).
2. The conditions become $C = DAB$ and $D = ABC$. Substituting the second into the first: $C = (ABC)(AB) = A^2B^2C$, so (for $C\ne0$) $A^2B^2=1$, i.e. $AB=\pm1$.
3. Choose $A=1, B=1$ (so $AB=1$). Then $D = ABC = C$, so set $C=D=5$ (any nonzero value works, consistent with both original relations: $C=DAB=5\cdot1\cdot1=5$ ✓, $D=ABC=1\cdot1\cdot5=5$ ✓).
4. Compute $S = ABCD = 1\cdot1\cdot5\cdot5 = 25$, so $S^2 = 625$.
5. Test each option against $S^2=625$: (A) $S=25 \ne 625$. (B) $BCD = 1\cdot5\cdot5=25 \ne 625$. (C) $S^T = S = 25$ (scalars are self-transpose) $\ne 625$.
6. (D) states $(S^T)^2 = (S^2)^T$. For scalars, $S^T=S=25$, so $(S^T)^2 = 625$; and $(S^2)^T = 625^T = 625$ (transpose of a scalar is itself). Both sides equal $625$ — holds.
7. Since options A, B, C all fail this valid test case, and D holds (as it must, being a universal transpose identity for any matrix — $(S^2)^T = (SS)^T = S^TS^T = (S^T)^2$), the answer is (D).

Common Mistakes

• Assuming the given relations must force $S$ to be idempotent ($S^2=S$) or that $S^2$ equals $S^T$ directly — neither survives a concrete numeric check.
• Overlooking that $(S^T)^2=(S^2)^T$ requires no special conditions at all — it's true for every square matrix, which is precisely why it's the safe, always-correct choice here.