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Problem 111

# Let $$(\mathrm{r} \times \mathrm{s})$$ denote a matrix with shape $(\mathrm{r} \times \mathrm{s})$. Find the shape of the following products if the product is defined (i) $$(2 \times 3)(3 \times 4)$$ (iii) $$(1 \times 2)(3 \times 1)$$ (v) $$(3 \times 4)(3 \times 4)$$ (ii) $$(4 \times 1)(1 \times 2)$$ (iv) $$(5 \times 2)(2 \times 3)$$ (vi) $$(2 \times 2)(2 \times 4)$$

Expert verified
(i) $$(2 \times 4)$$ (iii) Not defined (v) Not defined (ii) $$(4 \times 2)$$ (iv) $$(5 \times 3)$$ (vi) $$(2 \times 4)$$
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## Step 1: (i) $$(2 \times 3)(3 \times 4)$$

In this case, matrix A has 2 rows and 3 columns, and matrix B has 3 rows and 4 columns. Since the number of columns in A (3) is equal to the number of rows in B (3), we can multiply these matrices. The resulting matrix will have 2 rows (from A) and 4 columns (from B), so the shape is $$(2\times 4)$$.

## Step 2: (iii) $$(1 \times 2)(3 \times 1)$$

Matrix A has 1 row and 2 columns, while matrix B has 3 rows and 1 column. The number of columns in A (2) is not equal to the number of rows in B (3), so the product is not defined.

## Step 3: (v) $$(3 \times 4)(3 \times 4)$$

Both matrices A and B have 3 rows and 4 columns. The number of columns in A (4) is not equal to the number of rows in B (3), so the product is not defined.

## Step 4: (ii) $$(4 \times 1)(1 \times 2)$$

Matrix A has 4 rows and 1 column, while matrix B has 1 row and 2 columns. Since the number of columns in A (1) is equal to the number of rows in B (1), we can multiply these matrices. The resulting matrix will have 4 rows (from A) and 2 columns (from B), so the shape is $$(4\times 2)$$.

## Step 5: (iv) $$(5 \times 2)(2 \times 3)$$

Matrix A has 5 rows and 2 columns, while matrix B has 2 rows and 3 columns. Since the number of columns in A (2) is equal to the number of rows in B (2), we can multiply these matrices. The resulting matrix will have 5 rows (from A) and 3 columns (from B), so the shape is $$(5\times 3)$$.

## Step 6: (vi) $$(2 \times 2)(2 \times 4)$$

Matrix A has 2 rows and 2 columns, while matrix B has 2 rows and 4 columns. Since the number of columns in A (2) is equal to the number of rows in B (2), we can multiply these matrices. The resulting matrix will have 2 rows (from A) and 4 columns (from B), so the shape is $$(2\times 4)$$.

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