12th Grade

{
 "voice_prompt": "Pause briefly between single letters and numbers in formulas.",
 "manuscript": {
   "title": {
     "text": "How Do You Multiply Matrices by Scalars?",
     "audio": "How Do You Multiply Matrices by Scalars?"
   },
   "description": {
     "text": "A scalar is just a number you multiply each entry in the matrix by.",
     "audio": "A scalar is just a number you multiply each entry in the matrix by."
   },
   "scenes": [
     {
       "text": "Scalars are used to scale entire matrices. In matrix algebra, a scalar is a single number. It could be 2, negative 5, or zero point five. Scalar multiplication means multiplying every entry in the matrix by that number.",
       "latex": "a+b"
     },
     {
       "text": "For a 2-by-2 matrix A, with elements A, b, c, d, and a scalar, k, you multiply k by each element in A. That gives you a new matrix with the elements k a, k b, k c, and k d.",
       "latex": "A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}, \\quad kA = \\begin{bmatrix} ka & kb \\\\ kc & kd \\end{bmatrix}"
     },
     {
       "text": "For a numerical example, take matrix M with the numbers one, two, three, and four. The scalar is 3. To calculate 3 times M, you multiply 3 by each element.",
       "latex": "M = \\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\end{bmatrix} \\quad 3M"
     },
     {
       "text": "The calculation is 3 times 1, 3 times 2, 3 times 3, and 3 times 4. This gives you the resulting matrix: three, six, nine, twelve.",
       "latex": "3M = \\begin{bmatrix} 3 \\times 1 & 3 \\times 2 \\\\ 3 \\times 3 & 3 \\times 4 \\end{bmatrix} = \\begin{bmatrix} 3 & 6 \\\\ 9 & 12 \\end{bmatrix}"
     },
     {
       "text": "You can also use this to scale data. Imagine this Sales matrix shows how many Pizzas and Burgers were sold by two food trucks yesterday. The first row shows that Truck 1 sold 20 Pizzas and 30 Burgers. The second row shows that Truck 2 sold 25 Pizzas and 15 Burgers.",
       "latex": "\\text{Sales} = \\begin{bmatrix} 20 & 30 \\\\ 25 & 15 \\end{bmatrix}"
     },
     {
       "text": "If both trucks expect to sell twice as much tomorrow, you multiply the matrix by the scalar 2. The new Expected Sales matrix becomes forty, sixty, fifty, thirty.",
       "latex": "2 \\times \\begin{bmatrix} 20 & 30 \\\\ 25 & 15 \\end{bmatrix} = \\begin{bmatrix} 40 & 60 \\\\ 50 & 30 \\end{bmatrix}"
     }
   ],
   "outro": {
     "text": "A scalar is just a number you multiply each entry in the matrix by.",
     "audio": "A scalar is just a number you multiply each entry in the matrix by."
   }
 }
}

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