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How do you find the inverse of a matrix?

How do you find the inverse of a matrix?

{
 "voice_prompt": "Pause briefly between single letters like a, b, c. Do not read them as one continuous phrase.",
 "manuscript": {
   "title": {
     "text": "How do you find the inverse of a matrix?",
     "audio": "How do you find the inverse of a matrix?"
   },
   "description": {
     "text": "To find the inverse of a 2x2 matrix, you first calculate its determinant. If the determinant is not zero, you use a formula that involves swapping two elements, changing the sign of the other two, and dividing by the determinant.",
     "audio": "To find the inverse of a two-by-two matrix, you first calculate its determinant. If the determinant is not zero, you use a formula that involves swapping two elements, changing the sign of the other two, and dividing by the determinant."
   },
   "scenes": [
     {
       "text": "Before you can find the inverse of a 2-by-2 matrix, you must check its determinant. The formula is A, times D minus B times C. If the determinant is zero, the matrix has no inverse. The process stops here.",
       "latex": "A = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\det(A) = A \\times D - B \\times C"
     },
     {
       "text": "If the determinant isn’t zero, you can find the inverse using this formula, one over the determinant, times an adjusted matrix. To adjust the matrix, you swap the elements A, and D, and change the signs of B and C.",
       "latex": "A^{-1} = \\frac{1}{A \\times D - B \\times C} \\begin{bmatrix} D & -B \\\\ -C & A \\end{bmatrix}"
     },
     {
       "text": "Let's find the inverse of this matrix. First, you calculate the determinant, 4 times 6, minus 7 times 2, which equals 24 minus 14, which is 10. Since 10 isn’t zero, the inverse exists.",
       "latex": "A = \\begin{bmatrix} 4 & 7 \\\\ 2 & 6 \\end{bmatrix} \\det(A) = 4 \\times 6 - 7 \\times 2 = 10"
     },
     {
       "text": "Now you apply the formula. You take 1 over the determinant, which is 1 over 10. Then you adjust the matrix: swap the 4 and 6, and change the signs of 7 and 2.",
       "latex": "A^{-1} = \\frac{1}{10} \\begin{bmatrix} 6 & -7 \\\\ -2 & 4 \\end{bmatrix}"
     },
     {
       "text": "You multiply one over 10 into the adjusted Matrix using scalar multiplication. That gives you the final inverse matrix.",
       "latex": "A^{-1} = \\begin{bmatrix} 6/10 & -7/10 \\\\ -2/10 & 4/10 \\end{bmatrix} = \\begin{bmatrix} 0.6 & -0.7 \\\\ -0.2 & 0.4 \\end{bmatrix}"
     },
     {
       "text": "To check your answer, multiply the original matrix by the inverse. The answer should be the identity matrix. Start with the first row of the original matrix. Multiply it with each column of the inverse. First, 4 times 0.6 plus 7 times -0.2 gives 2.4 minus 1.4, which is 1. Then, 4 times -0.7 plus 7 times 0.4, which gives -2.8 plus 2.8, which is 0.",
       "latex": "4 \\times 0.6 + 7 \\times (-0.2) = 2.4 - 1.4 = 1 \\\\ 4 \\times (-0.7) + 7 \\times 0.4 = -2.8 + 2.8 = 0"
     },
     {
       "text": "Now for the second row. You get 2 times 0.6 plus 6 times -0.2, which equals 1.2 minus 1.2, which is 0. Then, 2 times -0.7 plus 6 times 0.4, which is -1.4 plus 2.4, which equals 1. You’ve just found the identity matrix. That confirms your inverse was correct.",
       "latex": "2 \\times 0.6 + 6 \\times (-0.2) = 1.2 - 1.2 = 0 \\\\ 2 \\times (-0.7) + 6 \\times 0.4 = -1.4 + 2.4 = 1"
     }
   ],
   "outro": {
     "text": "You’ve learned how to check if a 2-by-2 matrix has an inverse, and how to find it using the formula. Swap, change signs, divide by the determinant—and you’re done.",
     "audio": "You’ve learned how to check if a two-by-two matrix has an inverse, and how to find it using the formula. Swap, change signs, divide by the determinant—and you’re done."
   }
 }
}

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