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How to multiply matrices of appropriate dimensions?

How to multiply matrices of appropriate dimensions?

{
 "voice_prompt": "Pause slightly between single letters, variables, and matrix elements. Speak clearly when reading numbers or row-column pairs.",
 "manuscript": {
   "title": {
     "text": "How to Multiply Matrices of Appropriate Dimensions",
     "audio": "How to multiply matrices of appropriate dimensions"
   },
  "description": {
  "text": "To multiply matrices, you match rows from the first matrix with columns from the second. Each number in the result comes from multiplying and adding matching entries.",
  "audio": "To multiply matrices, you match rows from the first matrix with columns from the second. Each number in the result comes from multiplying and adding matching entries."
},
   "scenes": [
     {
       "text": "To multiply two matrices, say matrix A, by matrix B, there's a critical dimension rule, The number of columns in the first matrix, A, must be equal to the number of rows in the second matrix, B. If A is an m-by-n matrix and B is an n-by-p matrix, then they can be multiplied. The resulting matrix, C - equals A - times B, will be an m-by-p matrix.",
       "latex": "A_{m \\times n} \\times B_{n \\times p} = C_{m \\times p}"
     },
     {
       "text": "To find each element in matrix C, take a row from matrix A, and a column from matrix B. Multiply the matching entries, and add them up. For the element in row i, and column j, multiply row i, of A, by column j, of B, then sum the results. It sounds complex, but let’s break it down.",
       "latex": "C_{ij} = A_{i1}B_{1j} + A_{i2}B_{2j} + \\dots + A_{in}B_{nj}"
     },
     {
       "text": "Let A, be the matrix with elements one, two, three, four. Let B, be the matrix with elements five, six, seven, eight. Both are 2-by-2. The number of columns in A, equals the number of rows in B, so we can multiply them.",
       "latex": "A = \\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\end{bmatrix}, B = \\begin{bmatrix} 5 & 6 \\\\ 7 & 8 \\end{bmatrix}"
     },
     {
  "text": "Let’s multiply A, by B. One times five plus two times seven equals nineteen. One times six plus two times eight equals twenty-two. Three times five plus four times seven equals forty-three. Three times six plus four times eight equals fifty. So A - times B - equals the matrix, nineteen, twenty-two, forty-three, fifty.",
  "latex": "A \\times B = \\begin{bmatrix} 19 & 22 \\\\ 43 & 50 \\end{bmatrix}"
     },
     {
       "text": "Matrix multiplication is not commutative. A - times B is usually not the same as B times A. In fact, B times A, might not even be possible if the dimensions don't align. Always be careful about the order of multiplication.",
       "latex": "A \\times B \\neq B \\times A"
     },
     {
       "text": "Now let’s look at a different example. Let P, be a 3-by-2 matrix with rows: one zero, two one, and three two. Let Q, be a 2-by-3 matrix with rows: four five six, and seven eight nine. Can we multiply P times Q? Yes—we can. P has two columns and Q has two rows, so the multiplication is allowed. The result will be a 3-by-3 matrix.",
       "latex": "P = \\begin{bmatrix} 1 & 0 \\\\ 2 & 1 \\\\ 3 & 2 \\end{bmatrix}, Q = \\begin{bmatrix} 4 & 5 & 6 \\\\ 7 & 8 & 9 \\end{bmatrix}"
     },
     {
   "text": "Now let’s multiply P times Q. One times four plus zero times seven equals four. One times five plus zero times eight equals five. One times six plus zero times nine equals six. Two times four plus one times seven equals fifteen. Two times five plus one times eight equals eighteen. Two times six plus one times nine equals twenty-one. Three times four plus two times seven equals twenty-six. Three times five plus two times eight equals thirty-one. Three times six plus two times nine equals thirty-six. So P times Q equals the matrix with rows four five six, fifteen eighteen twenty-one, and twenty-six thirty-one thirty-six.",
   "latex": "P \\times Q = \\begin{bmatrix} 1\\times4+0\\times7 & 1\\times5+0\\times8 & 1\\times6+0\\times9 \\\\ 2\\times4+1\\times7 & 2\\times5+1\\times8 & 2\\times6+1\\times9 \\\\ 3\\times4+2\\times7 & 3\\times5+2\\times8 & 3\\times6+2\\times9 \\end{bmatrix} = \\begin{bmatrix} 4 & 5 & 6 \\\\ 15 & 18 & 21 \\\\ 26 & 31 & 36 \\end{bmatrix}"
     }
   ],
   "outro": {
     "text": "Matrix multiplication follows a row-by-column method and requires compatible dimensions. Though it may take several steps, it’s systematic—and essential in fields like computer graphics, data analysis, and engineering.",
     "audio": "Matrix multiplication follows a row-by-column method and requires compatible dimensions. Though it may take several steps, it’s systematic—and essential in fields like computer graphics, data analysis, and engineering."
   }
 }
}

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