(C) Given $A = [1\, 2\, 3]$ (a $1 \times 3$ matrix) and $B = \begin{bmatrix} -5 & 4 & 0 \\ 0 & 2 & -1 \\ 1 & -3 & 2 \end{bmatrix}$ (a $3 \times 3$ mat

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$\begin{bmatrix} -2 & -1 & 4 \end{bmatrix}$

(C) Given $A = [1\, 2\, 3]$ (a $1 \times 3$ matrix) and $B = \begin{bmatrix} -5 & 4 & 0 \\ 0 & 2 & -1 \\ 1 & -3 & 2 \end{bmatrix}$ (a $3 \times 3$ matrix). To find $AB$,we multiply the row of $A$ by each column of $B$: $AB = [1 \times (-5) + 2 \times 0 + 3 \times 1, \quad 1 \times 4 + 2 \times 2 + 3 \times (-3), \quad 1 \times 0 + 2 \times (-1) + 3 \times 2]$ $AB = [-5 + 0 + 3, \quad 4 + 4 - 9, \quad 0 - 2 + 6]$ $AB = [-2, \quad -1, \quad 4]$ Thus,$AB = [-2\, -1\, 4]$.

Class 12 Mathematics 3-4. Matrices and Determinants Types of matrices, Algebra of matrices

Easy

If $A = [1\, 2\, 3]$ and $B = \begin{bmatrix} -5 & 4 & 0 \\ 0 & 2 & -1 \\ 1 & -3 & 2 \end{bmatrix}$,then $AB = $

A
$\begin{bmatrix} -5 & 4 & 0 \\ 0 & 4 & -2 \\ 3 & -9 & 6 \end{bmatrix}$
B
$\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}$
C
$\begin{bmatrix} -2 & -1 & 4 \end{bmatrix}$
D
$\begin{bmatrix} -5 & 8 & 0 \\ 0 & 4 & -3 \\ 1 & -6 & 6 \end{bmatrix}$

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3-4. Matrices and Determinants

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Types of matrices, Algebra of matrices

If the matrix $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 2 \end{bmatrix}$, then $A^n = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ b & 0 & a \end{bmatrix}$, for $n \in N$, where:

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If $P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$,$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $Q = PAP^T$,then find $P^T Q^{2015} P$.

MediumJEE MAIN 2016

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Construct a $3 \times 4$ matrix,whose elements are given by $a_{i j}=\frac{1}{2}|-3 i+j|$.

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Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = \frac{1}{2}|i - 3j|$.

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Let $T$ and $U$ be the set of all orthogonal matrices of order $3$ over $\mathbb{R}$ and the set of all non-singular matrices of order $3$ over $\mathbb{R}$ respectively. Let $A = \{-1, 0, 1\}$. Then:

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If $A = [1\, 2\, 3]$ and $B = \begin{bmatrix} -5 & 4 & 0 \\ 0 & 2 & -1 \\ 1 & -3 & 2 \end{bmatrix}$,then $AB = $

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If the matrix $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 2 \end{bmatrix}$, then $A^n = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ b & 0 & a \end{bmatrix}$, for $n \in N$, where:

If $P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$,$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $Q = PAP^T$,then find $P^T Q^{2015} P$.

Construct a $3 \times 4$ matrix,whose elements are given by $a_{i j}=\frac{1}{2}|-3 i+j|$.

Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = \frac{1}{2}|i - 3j|$.

Let $T$ and $U$ be the set of all orthogonal matrices of order $3$ over $\mathbb{R}$ and the set of all non-singular matrices of order $3$ over $\mathbb{R}$ respectively. Let $A = \{-1, 0, 1\}$. Then:

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