12th Grade
How to add and subtract matrices: the dimension rule
{
"voice_prompt": "Pause briefly between single letters like a, b, c, and between operators like plus and times. Do not read them as one continuous phrase.",
"manuscript": {
"title": {
"text": "How to Add and Subtract Matrices: The Dimension Rule.",
"audio": "How to Add and Subtract Matrices: The Dimension Rule."
},
"description": {
"text": "You can only add or subtract matrices if they have the same dimensions. Then, you just combine corresponding elements.",
"audio": "You can only add or subtract matrices if they have the same dimensions. Then, you just combine corresponding elements."
},
"scenes": [
{
"text": "Before you can add or subtract matrices, they must have exactly the same number of rows and the same number of columns. We say they have the same dimensions. If the sizes don't match, there's no way to pair them up.",
"latex": "\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} + \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix}"
},
{
"text": "For example, a 2-by-2 matrix can be added to another 2-by-2 matrix, but not to a 2-by-3 matrix.",
"latex": "\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} + \\begin{bmatrix} e & f & x \\\\ g & h & y \\end{bmatrix}"
},
{
"text": "To add two matrices, A and B, that have the same dimensions, you simply add their corresponding elements.",
"latex": "A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}, B = \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix}"
},
{
"text": "If matrix A, has the elements A, b, c and d, and matrix B has the elements e, f, g and h, then A, plus B equals A, plus e, b plus f, c plus g and d plus h.",
"latex": "A+B = \\begin{bmatrix} a+e & b+f \\\\ c+g & d+h \\end{bmatrix}"
},
{
"text": "Let's add two 2-by-2 matrices. Matrix M has the elements one, two, three and four and matrix N has the elements five, six, seven and eight.",
"latex": "M = \\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\end{bmatrix}, N = \\begin{bmatrix} 5 & 6 \\\\ 7 & 8 \\end{bmatrix}"
},
{
"text": "Then M plus N equals one plus five, two plus six, three plus seven and four plus eight. That gives us six, eight, ten and twelve.",
"latex": "M+N = \\begin{bmatrix} 1+5 & 2+6 \\\\ 3+7 & 4+8 \\end{bmatrix} = \\begin{bmatrix} 6 & 8 \\\\ 10 & 12 \\end{bmatrix}"
},
{
"text": "Subtracting matrices works exactly the same way! Again, they must have the same dimensions. You just subtract the corresponding elements. For example, if matrix A has the elements a, b, c and d, and matrix B has the elements e, f, g and h,",
"latex": "A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}, B = \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix}"
},
{
"text": "then A, minus B equals A, minus e, b minus f, c minus g and d minus h.",
"latex": "A-B = \\begin{bmatrix} a-e & b-f \\\\ c-g & d-h \\end{bmatrix}"
},
{
"text": "Let's subtract our matrices M and N. M has the elements one, two, three and four. N has the elements five, six, seven and eight.",
"latex": "M = \\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\end{bmatrix}, N = \\begin{bmatrix} 5 & 6 \\\\ 7 & 8 \\end{bmatrix}"
},
{
"text": "Then M minus N equals one minus five, two minus six, three minus seven and four minus eight. That gives negative four in every position.",
"latex": "M-N = \\begin{bmatrix} 1-5 & 2-6 \\\\ 3-7 & 4-8 \\end{bmatrix} = \\begin{bmatrix} -4 & -4 \\\\ -4 & -4 \\end{bmatrix}"
},
{
"text": "Imagine matrix J shows sales for January and matrix F for February. J has the numbers ten, fifteen, twenty and twenty-five. F has twelve, eighteen, twenty-two and twenty. ",
"latex": "J = \\begin{bmatrix} 10 & 15 \\\\ 20 & 25 \\end{bmatrix}, F = \\begin{bmatrix} 12 & 18 \\\\ 22 & 20 \\end{bmatrix}"
},
{
"text": "To find the total sales, just add the matching numbers. Ten plus twelve, fifteen plus eighteen, 20 plus 22 and 25 plus 20. The total becomes twenty-two, thirty-three, forty-two and forty-five.",
"latex": "J+F = \\begin{bmatrix} 10+12 & 15+18 \\\\ 20+22 & 25+20 \\end{bmatrix} = \\begin{bmatrix} 22 & 33 \\\\ 42 & 45 \\end{bmatrix}"
}
],
"outro": {
"text": "Adding and subtracting matrices is all about working with corresponding elements, but only if the matrices are the same size! Always check dimensions first.",
"audio": "Adding and subtracting matrices is all about working with corresponding elements, but only if the matrices are the same size! Always check dimensions first."
}
}
}