For the following exercises, use the matrices below to...

For the following exercises, use the matrices below to perform matrix multiplication. $$ A=\left[\begin{array}{rr}{-1} & {5} \\ {3} & {2}\end{array}\right], B=\left[\begin{array}{rrr}{3} & {6} & {4} \\ {-8} & {0} & {12}\end{array}\right], C=\left[\begin{array}{rr}{4} & {10} \\ {-2} & {6} \\ {5} & {9}\end{array}\right], D=\left[\begin{array}{rrr}{2} & {-3} & {12} \\ {9} & {3} & {1} \\ {0} & {8} & {-10}\end{array}\right] $$ $$ A B $$

For the following exercises, use the matrices below to perform matrix multiplication.
$$
A=\left[\begin{array}{rr}{-1} & {5} \\ {3} & {2}\end{array}\right], B=\left[\begin{array}{rrr}{3} & {6} & {4} \\ {-8} & {0} & {12}\end{array}\right], C=\left[\begin{array}{rr}{4} & {10} \\ {-2} & {6} \\ {5} & {9}\end{array}\right], D=\left[\begin{array}{rrr}{2} & {-3} & {12} \\ {9} & {3} & {1} \\ {0} & {8} & {-10}\end{array}\right]
$$
$$
A B
$$

Key Concepts

Resultant Matrix

The resultant matrix obtained from multiplication has dimensions determined by the number of rows in the first matrix and the number of columns in the second matrix. Each entry in this matrix is calculated as the dot product of the corresponding row and column from the original matrices.

Dimensions Compatibility

Before multiplying two matrices, it is important to check that the number of columns in the first matrix equals the number of rows in the second matrix. This compatibility ensures that each element in the resulting matrix can be computed by taking the dot product of a row from the first matrix and a column from the second matrix.

Dot Product

The dot product is a calculation performed between two sequences of numbers, where corresponding elements are multiplied and then summed together. In matrix multiplication, this operation is used to compute each entry of the product matrix by combining a row from the first matrix with a column from the second matrix.

Matrix

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is a fundamental object in linear algebra used to represent and solve systems of linear equations, perform transformations, and more.

Matrix Multiplication

Matrix multiplication is an operation in which two matrices are multiplied together to produce a new matrix. This operation involves taking rows from the first matrix and columns from the second, multiplying corresponding entries, and summing these products. The process is not commutative, meaning the order of multiplication matters.

Transcript

00:02 Okay, so when we're doing matrix multiplication, we want to check that the inner dimensions of the matrices involved are the same.

00:17 So we have a and b involved.

00:21 So the dimensions for a are two rows and two columns, two by two.

00:27 And the dimensions for b are two rows and three columns, so two by three.

00:34 And so what the inner dimensions are basically the column of a and the row of b which are two so they're the same and this property allows us to engage in the operation of matrix multiplication okay so that means that we can proceed into doing matrix multiplication so we have a a plus b a times b is equal two the entries of a are negative one five three and two and the entries for b are three negative eight six zero four so okay and now we want to do is take a single a row so this row and then we multiply this row by a column okay so what we have is negative one times three and we sum them so plus five times negative eight and then we take the row again and multiply by the next column so we have negative one times six plus five times zero and then we take the same row from here and then we multiply by the third column.

02:17 So we have minus 1 times 4 plus 5 times 12.

02:25 And now we go to the second row.

02:29 So we take this row on the bottom and then we do the same thing basically.

02:34 So which is 3 times 3 plus 2 times negative 8.

02:44 And then we have 3 times 6.

02:46 Plus two times zero...

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